1.1 General Group Theory

KNOW BASIC DEFINITIONS: groups (subgroups, normal and characteristic subgroups, quotients; simple, abelian, solvable, nilpotent groups; commutators, normalizers, centralizers); order, index; homomorphisms (iso, mono, epi, auto, inner); group actions, transitive, orbits, stabilizers, fixed points.

Know the following theorems: the three isomorphism theorems, the correspondence theorem (for subgroups of $G / N$ ); Lagrange’s theorem, the product formula for $|A B|$ ; the fundamental theorem for finitely generated abelian groups (see Section 1.3 below); automorphisms of cyclic groups; theorems about $p$ -groups (nilpotency; groups of order $p^{2}$ are abelian); Cauchy’s theorem; the Sylow theorems (see Section 1.2 below).

Know solvability and nilpotency: If $N \triangleleft G$ then $G$ is solvable if and only if $G / N$ and $N$ are solvable. Solvability chain of derived groups $G^{(n)}$ , where $G^{(0)}:=G, G^{(n+1)}:=\left[G^{(n)}, G^{(n)}\right]$ . The (lower) central series $G^{[n]}$ , where $G^{[0]}:=G, G^{[n+1]}:=\left[G^{[n]}, G\right]$ . Subgroups and quotients of solvable/nilpotent groups are solvable/nilpotent. A nilpotent group has non-trivial center $Z(G)$ . If $G / N$ is nilpotent for $N \leq Z(G)$ , then also $G$ is nilpotent. If $H$ is a proper subgroup of a nilpotent group $G$ , then $N_{G}(H) \neq H$ .

Know the following constructions: direct and semidirect products, "inner" and "outer" version, characterizations when a group is a (semi)direct product of two subgroups.

Know the following classes of examples: cyclic groups and their subgroup structure; symmetric groups $S_{n}$ , cycle decompositions, conjugation of cycles, the sign function, alternating groups $A_{n}$ ; dihedral groups $D_{2 n}$ ; the quaternion group; general linear groups $G L_{n}(F)$ ( $F$ a field) and related groups (like $S L_{n}(F)$ ).

Know Group Actions: If a group $G$ acts on a set $S$ then

$S$ is the disjoint union of orbits.
$|\operatorname{Orbit}(s)|=\left[G: G_{s}\right]$ .
$\mid$ Conjugacy class $(x) \mid=\left[G: C_{G}(x)\right]$ .
Class equation:

$|G|=|Z(G)|+\sum_{i=1}^{m}\left[G: C_{G}\left(x_{i}\right)\right]$

where $x_{1}, \ldots, x_{m}$ are representatives of all distinct conjugacy classes in $G$ consisting of more than one element.
5. If $G$ is $p$ -group and $S$ is finite, then $\mid$ fixed points $|\equiv| S \mid(\bmod p)$ .

AVOID THE FOLLOWING MISTAKES (incomplete list!): being normal is not a transitive relation; $\operatorname{ord}(g h)$ is usually (for nonabelian groups) not related to $\operatorname{ord}(g)$ and $\operatorname{ord}(h)$ ; a subgroup of $G \times H$ need not be of the form $G_{1} \times H_{1}$ with $G_{1} \leq G$ and $H_{1} \leq H$ ; an abelian subgroup $H \leq G$ need not be contained in $Z(G)$ ; if $N$ and $G / N$ are nilpotent, $G$ need not be nilpotent (as opposed to solvable groups!); if a prime power $p^{n}$ divides $|G|$ , then $G$ need not have an element of order $p^{n}$ (but it has one of order $p$ by Cauchy’s theorem); if $m$ divides $|G|$ , then there need not exist a subgroup of $G$ of order $m$ (but it must exist if $G$ is abelian!) ...

(April $77 \# 5$ ) (a) Show the alternating group $A_{n}$ is normal in $S_{n}$ .
(b) Show $A_{n}$ is generated by all 3 -cycles ( $12 k$ ) for $k=3,4, \ldots, n$ .
(c) Show any normal subgroup of $A_{n}$ which contains a 3 -cycle must be all of $A_{n}$ .
(May 78 #1) Name a nonabelian simple group.
(May $78 \# 2$ ) If a finite $p$ -group $G$ acts linearly on a finite-dimensional vector space over $\mathbb{F}_{p}$ , show $G$ has a nonzero fixed point.
(Jan 79 #8) Do the elements of finite order in a group always form a subgroup?
(March $83 \# 3$ ) Show that $S_{8}$ contains a subgroup $H$ of order 15 , but $S_{n}$ for $n<8$ doesn’t.
(Feb $84 \# 7$ ) If $H, K$ are subgroups of $G$ with $H a=K b$ for some $a, b$ in $G$ , prove $H=K$ . What can you say if $a H=K b$ ?
(Sep $86 \# 6$ ) For what $n$ is $S_{n} \longrightarrow \operatorname{Aut}\left(S_{n}\right)$ (via $g \longrightarrow \kappa_{g}$ conjugation by $g$ ) a monomorphism?
(Jan $87 \# 2$ ) If $G$ is infinite but some nontrivial element $x \neq 1$ has only a finite number of conjugates, show $G$ is not simple.
(Aug 89 #1) State the class equation for a finite group, and use it to show $Z(G)>1$ for a nontrivial $p$ -group $G$ , then prove $G$ is nilpotent.
(Jan $89 \# 2$ ) (a) If $G$ has a normal subgroup $N$ with $G / N=\mathbb{Z}$ , show for all $n \neq 0$ there is a normal subgroup $N_{n}$ with $G / N_{n}=\mathbb{Z}_{n}$ .
(b) If all proper factor groups $G / N$ of $G$ are finite, must $G$ be finite?
(May $89 \# 5$ ) If $H, K$ are subgroups of $G$ show $G$ is a disjoint union of double cosets $H g K$ .
(May $90 \# 1$ ) If $G$ has no nontrivial automorphisms, prove it has order 1 or 2 .
(May $92 \# 3$ ) Prove $V=\{1,(12)(34),(13)(24),(14)(23)\}$ is a normal subgroup of the symmetric group $S_{4}$ on 4 symbols, and that $S_{4} / V \cong S_{3}$ .
(May 92 #5) Use the subgroup structure of the cyclic group of order $n \geq 1$ to show that $n=\sum_{d \mid n} \phi(d)$ , where the Euler $\phi$ -function $\phi(n)$ is the number of integers $1 \leq k \leq n$ which are relatively prime to $n$ .
(Sep 93 #7) A well-known puzzle has tiles numbered 1 to 15 in 4 rows of 4 each, with the $(4,4)$ square empty. An allowable move consists of sliding a tile adjacent to the empty square horizontally or vertically into the empty square. If a sequence of moves ends up with the empty square back at $(4,4)$ , prove the resulting permutation $\pi$ of the numbers 1 to 15 belongs to $A_{15}$ .
(Jan 94 #6) (a) Explain why the inner automorphism group of the alternating group $A_{n}$ is isomorphic to $A_{n}$ for $n \geq 4$ .
(b) Prove that for $n \geq 3, A_{n}$ has outer (:= not inner) automorphisms.
(Aug $95 \# 1$ ) Let $G$ be a finite group of permutations of a finite set $X$ . For $x \in X$ let $G_{x}=\operatorname{Stab}(x)=\{g \in G \mid g x=x\}$ . If $|X|=\left[G: G_{x}\right]$ for some $x \in X$ , show the same holds for all $x \in X$ .
(Jan 95 #5) Give definitions of the terms "maximal subgroup" and "minimal subgroup" ; it is not assumed that you have seen these terms previously. Then from your definitions, prove the following facts:
(a) A minimal subgroup must be cyclic of prime order.
(b) If a subgroup has prime index, it is a maximal subgroup.
(c) If a subgroup is both maximal and normal, it has prime index.
(d) A subgroup of an abelian group is maximal if and only if it has prime index.
(e) Find all maximal and minimal subgroups of $\mathbb{Z}$ .
(Aug 95 #7) Find the order of the group $G L_{n}\left(\mathbb{Z}_{p}\right)$ and describe one of its $p$ -Sylow subgroups.
(Aug $96 \# 1$ ) A Hall subgroup $H$ of a finite group $G$ is a subgroup whose order and index are relatively prime. Use isomorphism theorems to prove that if $N$ is a normal subgroup of $G$ and $H$ is a Hall subgroup of $G$ , then $H N / N$ is a Hall subgroup of $G / N$ , and $H \cap N$ is a Hall subgroup of $N$ .
(Aug $97 \# 2$ ) (a) Prove that if $G$ is a finite group with exactly two conjugacy classes of elements, then $|G|=2$ .
(b) If $G$ has exactly three conjugacy classes of elements, show that $|G|$ involves at most two primes.
(c) There are, in fact, only two finite groups with exactly three conjugacy classes of elements. Can you guess which ones they are?
(Aug $98 \# 2$ ) Let $G L_{2}(p)$ for a prime $p$ denote the group of invertible $2 \times 2$ matrices over the finite field $\mathbb{F}_{p}$ of $p$ elements.
(a) Find the order $n$ of the group $G L_{2}(p)$ .
(b) For $\lambda$ in $\mathbb{F}_{p}$ , and $B:=\left(\begin{array}{cc}\lambda & 1 \\ 0 & \lambda\end{array}\right)$ , find the order $m$ of the subgroup

$G_{\lambda}:=\left\{A \in G L_{2}(p) \mid A B A^{-1}=B\right\}$

(c) Find how many $2 \times 2$ matrices over $\mathbb{F}_{p}$ are similar to the matrix $B$ . (Hint: express it in terms of $m$ and $n$ .)
23. (Aug $99 \# 2$ ) Let $G$ be a finite $p$ -group for a prime $p$ having a unique subgroup $G_{p}$ of order $p$ . (The quaternion group is such a group, with $p=2$ and $G_{2}=\{1,-1\}$ .)
(a) Show that $G_{p}$ is invariant under all endomorphisms of $G, f\left(G_{p}\right) \subseteq G_{p}$ for all homomorphisms $f: G \longrightarrow G$ .
(b) Show that $G_{p}$ "needs room" in order to act: whenever $G$ acts on a finite set $S$ of size $|S|<|G|$ , the subgroup $G_{p}$ acts trivially. Conclude that $G$ can only act faithfully on sets of size $\geq|G|$ .
24. (Jan 00 #4) (a) If $G$ is a group containing a cyclic normal subgroup $N$ , show that $g n=n g$ for all $n$ in $N$ and all $g$ in the commutator subgroup of $G$ .
(b) Suppose that $N_{1}, N_{2}, N_{3}$ are three normal subgroups of a group $G$ with the properties that for distinct $i, j$ always $N_{i} \cap N_{j}=1, N_{i} N_{j}=G$ . Show that all three subgroups $N_{i}$ are isomorphic, and that $G$ is abelian. Give an example of such an abelian group of order 4.
25. (Aug $01 \# 1$ ) If $\phi: G_{1} \rightarrow G_{2}$ is a homomorphism of groups, and $N_{1} \triangleleft G_{1}, N_{2} \triangleleft G_{2}$ are two normal subgroups, show that the map $\bar{\phi}$ given on cosets by $\bar{\phi}\left(x N_{1}\right)=\phi(x) N_{2}$ is a welldefined homomorphism $\phi: G_{1} / N_{1} \rightarrow G_{2} / N_{2}$ of quotient groups if and only if the original homomorphism satisfies $\phi\left(N_{1}\right) \subseteq N_{2}$ .
26. (Aug $02 \# 3$ ) Let $G$ be a group. Show that if, for any three elements $x, y, z \in G$ , at least two of them commute, then $G$ itself is abelian. (Hint. Use centralizers.)
27. (Aug 02 #4) Show that a finite group is noncommutative if and only if it has an irreducible complex representation of dimension $>1$ .
28. (Aug $02 \# 7$ ) Let $p$ be a prime $>2$ .
(a) Show that in the symmetric group $S_{p}$ , any two elements of order $p$ are conjugate.
(b) Exhibit two nonconjugate elements of order $p$ in the symmetric group $S_{p^{2}}$ .
(c) Are there nonconjugate elements of order $p$ in the group $G L_{2}(\mathbb{C})$ ?
(d) Can you find an element of order $p$ in $G L_{2}(\mathbb{R})$ ?
29. (Jan $04 \# 1$ ) Show that the group $G_{1}$ of all real numbers for addition is isomorphic to the group $G_{2}$ of all positive real numbers for multiplication. Furthermore, show that the group $H_{1}$ of all rational numbers is not isomorphic to the group $H_{2}$ of all positive rational numbers for multiplication.
30. (Jan $04 \# 2$ ) Let $G$ be a group such that there exists a surjective group homomorphism $G \rightarrow \mathbb{Z}$ . Prove that for any subgroup of finite index $H \leq G$ there also exists a surjective group homomorphism $H \rightarrow \mathbb{Z}$ .
31. (Jan $04 \# 5$ ) Let $G=\left\{g_{1}, \ldots, g_{n}\right\}$ be a finite abelian group. Show that the product $P:= g_{1} \ldots g_{n}$ is of order 1 or 2 .
32. (Jan 04 #8) Let $S$ be a finite set acted upon by a finite group $G$ . Denote by $\mathbb{C}(S)=\{f: S \rightarrow \mathbb{C}\}$ the space of functions on $S$ .
(a) Show that the map $G \times \mathbb{C}(S) \rightarrow \mathbb{C}(S),(x, f) \mapsto x f$ , defines a $G$ -action on $\mathbb{C}(S)$ . Here $x f \in \mathbb{C}(S)$ is defined by $x f(s)=f\left(x^{-1} s\right)$ .
(b) Show that the dimension of the subspace of $G$ -fixed points in $\mathbb{C}(S)$ is equal to the number of $G$ -orbits in $S$ .
33. (Aug $04 \# 1$ ) Assume that the group $G$ is a direct product of two finite subgroups $A$ and $B$ , $G=A \times B$ , and that $|A|,|B|$ are relatively prime. Show that $H=(H \cap A) \times(H \cap B)$ for any subgroup $H$ of $G$ .
34. (Jan $05 \# 1$ ) Write down the class equation for the dihedral group $D_{20}$ of order 20.

No proof required but you should identify the terms in your equation.
35. (Aug 05 #2) Let G be finitely generated group.
(a) If $H$ is a finite group, show that there exist only finitely many group homomorphisms from $G$ to $H$ .
(b) If $n$ is a given natural number, show that there exist only finitely many normal subgroups $N$ of $G$ with $[G: N]=n$ .
36. (Aug $06 \# 2$ ) Let $G$ be a finite group wherein any two conjugate elements commute. Prove that $G$ is solvable. (More is true: $G$ actually has to be nilpotent; that is, however, a little harder to show.) There is partial credit for proving that $G$ is not simple unless it is abelian.
37. (Jan 08 #1) Show that a group $G$ will have outer automorphisms (automorphisms which are not inner) if it can be properly imbedded as a normal subgroup $G \triangleleft G^{\prime}$ of a group in such a way that $G \cdot \operatorname{Centralizer}_{G^{\prime}}(G) \neq G^{\prime}$ .
(b) Show that $A_{n}$ has outer automorphisms whenever $n \geq 4$ .
(c) Explain the mantra "Every automorphism of $G$ is inner, somewhere."
38. (Aug 08 #7) The goal of this problem is to prove that all automorphisms $\varphi$ of $S_{5}$ are inner. a. Given that the transpositions $(i, j)(i<j)$ generate $S_{n}$ , prove that the adjacent transpositions $\tau_{i}:=(i, i+1) \in S_{n}$ for $1 \leq i \leq n-1$ generate $S_{n}$ .
b. Prove that all automorphisms of $S_{n}$ leave $A_{n}$ invariant, $\varphi\left(A_{n}\right)=A_{n}$ .
c. Prove that the elements of order 2 in $S_{5}$ are precisely all single and double transpositions $(i, j)$ and $(i, j)(k, \ell)$ for distinct $i, j, k, \ell$ , then use part (b) to show that $\varphi\left(\tau_{i}\right)$ is a single transposition for every automorphism $\varphi \in \operatorname{Aut}\left(S_{5}\right)$ .
d. Prove that every automorphism $\varphi \in \operatorname{Aut}\left(S_{5}\right)$ is an inner automorphism. [Hint: count possibilities for $\varphi\left(\tau_{i}\right)$ to get an upper bound on the number of automorphisms of $S_{5}$ ].
39. (Aug $09 \# 2$ ) Let $G$ be a finite group and let $H$ and $K$ be subgroups of $G$ . For each $x \in G$ define $H x K=\{h x k: h \in H, k \in K\}$ .
(a) Prove that for any $x, y \in G$ either $H x K=H y K$ or $H x K \cap H y K=\emptyset$ .
(b) Prove that $|H x K|=|H||K| /\left|H \cap x K x^{-1}\right|$ . Hint: Use group actions: either a suitable action of $H \times K$ on $G$ or a suitable action of $H$ on $G / K$ .
40. (Aug $09 \# 8$ ) Let $F$ be a field. Prove that the additive and multiplicative groups of $F$ cannot be isomorphic. Hint: Look at the orders of elements in both groups.
41. (Aug $10 \# 2$ ) Let $G$ be a group. A subgroup $H$ of $G$ will be called essential if $H \cap K \neq\{1\}$ for every non-trivial subgroup $K$ of $G$ .
(a) Let $p$ be a prime and $k \geq 2$ . Prove that the group $\mathbb{Z} / p^{k} \mathbb{Z}$ has a proper essential subgroup.
(b) Assume that $H_{1}$ is an essential subgroup of $G_{1}$ and $H_{2}$ is an essential subgroup of $G_{2}$ . Prove that $H_{1} \times H_{2}$ is an essential subgroup of $G_{1} \times G_{2}$ .
(c) Let $G$ be a finite abelian group. Prove that $G$ does not have a proper essential subgroup if and only if $G$ is a direct product of groups of prime order.
42. (Aug 11 # 2) The following question concerns symmetric groups. You can assume as given the fact that any permutation in $S_{n}$ can be written (uniquely up to order) as a (commuting) product of disjoint cycles (of varying lengths). Otherwise, your argument should be selfcontained.
(a) For $n \geq 2$ , show that the symmetric group $S_{n}$ is generated by the transpositions (i, j), $1 \leq i<j \leq n$ .
(b) For $n \geq 3$ , show that the alternating group $A_{n}$ is generated by the 3 -cycles ( $1,2, \mathrm{i}$ ), $2<i \leq n$ .
(c) Let $H$ be a subgroup of a group $G$ of index $n$ . Show that $G$ has a normal subgroup $N$ which is contained in $H$ and which has index $\leq n!$ .
43. (Aug 11 # 6a) Suppose that $G=C_{p} \times \ldots \times C_{p}$ is a direct product of n copies of the cyclic group $C_{p}$ of order p. How many subgroups does $G$ have of order $p$ ? How many does it have of order $p^{n-1}$ ? Explain.
44. (Jan $12 \# 2$ ) Let $G$ be a subgroup of the symmetric group $S_{n}$ for some integer $n>1$ . Assume that $G$ acts transitively on $\mathbf{n}:=\{1,2, \ldots, n\}$ , that is, for any $i, j \in \mathbf{n}$ there exists $g \in G$ s.t. $g(i)=j$ .
A partition of $\mathbf{n}$ is a decomposition $\mathbf{n}=X_{1} \cup \ldots \cup X_{m}$ into a disjoint union of nonempty subsets. There are two trivial partitions: $\mathbf{n}=\mathbf{n}$ and $\mathbf{n}=X_{1} \cup \ldots \cup X_{n}$ (so each $X_{i}$ has just one element). Otherwise the partition is said to be nontrivial. The group $G$ is called imprimitive if there is a nontrivial partition $\mathbf{n}=X_{1} \cup \ldots \cup X_{m}$ such that, for $g \in G$ and $1 \leq i \leq m, g\left(X_{i}\right)=X_{j}$ for some $j$ . (That is, $G$ permutes the partition members among themselves.) The set $\left\{X_{i}\right\}$ is called a system of imprimitivity for the action of G on $\mathbf{n}$ . The group G is called primitive if it is not imprimitive.
(a) Let $n=6$ and consider the cyclic subgroup $G:=\langle(1,2,3,4,5,6)\rangle$ of $S_{6}$ . There are two non-trivial systems of imprimitivity for the action of $G$ on $\mathbf{n}$ . Find them.
(b) Prove that if $X_{1} \cup \ldots \cup X_{m}$ is a system of imprimitivity for the action of $G$ on $\mathbf{n}$ , then all subsets $X_{i}$ have the same size $n / m$ .
(c) $G$ is said to be doubly transitive if given elements $a, b, c, d \in \mathbf{n}$ , with $a \neq b$ and $c \neq d$ , there exists $g \in G$ such that $g(a)=c$ and $g(b)=d$ . Show that a doubly transitive group $G$
is primitive.
(d) Show that if $n \geq 3$ , the alternating subgroup $G=A_{n}$ of $S_{n}$ is primitive.
45. (Aug $12 \# 1$ ) For a positive integer $n$ , denote by $S_{n}$ the symmetric group on $\{1,2, \ldots, n\}$ . Let $p>2$ be a prime number.
(a) Give an example of a non-cyclic group of order $2 p$ .
(b) Find the smallest $n$ for which $S_{n}$ contains a cyclic subgroup of order $2 p$ .
(c) Find the smallest $n$ for which $S_{n}$ contains some subgroup of order $2 p$ .

In both (b) and (c), if $n$ is your answer, explain clearly why $S_{n}$ contains a desired subgroup and why $S_{m}$ for $m<n$ does not contain such subgroup.
46. (Jan $13 \# 1$ ) Let $p$ be a prime and let $S_{2 p}$ denote the symmetric group on $2 p$ elements.
(a) Find the order of a $p$ -Sylow subgroup of $S_{2 p}$ .
(b) Describe explicitly a $p$ -Sylow subgroup of $S_{2 p}$ (providing a generating set counts as explicit description, but make sure to prove that your subgroup is indeed $p$ -Sylow).
(c) Consider the set of elements of order $p$ in $S_{2 p}$ - clearly, it is a union of conjugacy classes. How many conjugacy classes does it consist of?
(d) Now consider the set of elements of order $p$ in the alternating group $A_{2 p}$ . How many conjugacy classes ( of $A_{2 p}$ ) does it consist of? Make sure to justify your answer.

Hint: Distinguish between the cases $p=2$ and $p>2$ .
47. (Aug $13 \# 1$ ) Let $p$ be an odd prime and $G$ a nonabelian group of order $p^{3}$ .
(a) Prove that $|Z(G)|=p$
(b) Prove that $Z(G)=[G, G]$ .
48. (Aug $13 \# 3$ ) If $G$ is a group, then there is a natural action of $\Sigma_{n}$ on $G^{\times n}$ given by permuting the factors. Define the wreath product $G \geq \Sigma_{n}$ to be

$G \succ \Sigma_{n}=G^{n} \rtimes \Sigma_{n}$

using this action of $\Sigma_{n}$ on $G^{n}$ .
(a) ( 3 points) If $X$ is a $G$ -set, show that $X^{n}$ is naturally a $G$ ? $\Sigma_{n}$ set by combining two actions: $G^{n}$ on $X^{n}$ via

$\left(g_{1}, \ldots, g_{n}\right) \cdot\left(x_{1}, \ldots, x_{n}\right)=\left(g_{1} x_{1}, \ldots, g_{n} x_{n}\right)$

for $\left(g_{1}, \ldots, g_{n}\right) \in G^{n}$ and $\left(x_{1}, \ldots, x_{n}\right) \in X^{n}$ , and $\Sigma_{n}$ on $X^{n}$ via

$\sigma \cdot\left(x_{1}, \ldots, x_{n}\right)=\left(x_{\sigma^{-1}(1)}, \ldots, x_{\sigma^{-1}(n)}\right),$

where $\sigma \in \Sigma_{n}$ .
(b) Show that $\Sigma_{n} \prec \Sigma_{m}$ embeds into $\Sigma_{n m}$ .
(c) Identify $\Sigma_{2} \zeta \Sigma_{2}$ with a more familiar group
(d) Determine the order of $G \succ \Sigma_{n}$ as a function of the orders of $G$ and $n$
(e) Bonus: Determine (no proof needed) the $p$ -Sylow subgroup of $\Sigma_{p^{k+1}}$ as a function of $k$ . Provide no more than a sentence of justification.
49. (Jan 14 #8) Use the semidirect product constructions to classify the groups of order 44. (Hint: start by analyzing Sylow subgroup structures.)
50. (Aug $14 \# 1$ ) Let G be a finite nilpotent group, $\mathrm{Z}(\mathrm{G})$ its center and p a prime number. Prove that p divides $|G|$ if and only if $p$ divides $|Z(G)|$ .
51. (Aug $14 \# 3$ ) Let p be a a prime number.
(a) Determine the order of the automorphism group of $\mathbb{Z}_{p} \times \mathbb{Z}_{p}$ .
(b) Prove that there exists a non-abelian group of order $p^{3}$ .
52. (Aug 14 #6) (a) Compute the order of the group $G$ of rigid motions of a regular octahedron $O$ .
(b) The group $G$ acts on the set of vertices of $O$ . Describe the stabilizer of a vertex of $O$ .
53. (Aug $15 \# 3$ ) (a) Let $G$ be a group of order $2 n$ , where $n$ is odd and $n>1$ . Prove that $G$ cannot be simple.
(Hint: consider elements of order 2 in the regular representation of $G$ in $S_{2 n}$ .)
(b) Let $G=\mathbb{Z}_{n}^{*}$ denote the group of units in $\mathbb{Z}_{n}$ . Find all integers $n$ such that $x^{2}=1$ for all $x \in G$ .
54. (Jan $16 \# 2$ ) Let $G$ be an infinite group and let $H$ be a subgroup of finite index. Prove that there exists a subgroup $K$ of $H$ such that $K$ has finite index in $G$ and such that $K$ is normal in $G$ .
55. (Jan $16 \# 4$ ) Let $G=S_{n}$ be the symmetric group on $n$ elements, and let $\sigma=(123 \ldots n)$ be an $n$ -cycle. Let $K$ be the cyclic subgroup generated by $\sigma$ . Prove that the order of the normalizer of $K$ , i.e., the order of the subgroup $H=\left\{x \in S_{n} \mid x^{-1} \sigma x \in K\right\}$ , is exactly $n \cdot \phi(n)$ , where $\phi(n)$ is the Euler $\phi$ -function. (Recall that $\phi(n)$ is the number of positive integers less than $n$ and relatively prime to $n$ .)
56. (Aug $16 \# 2$ ) What is the smallest integer $m \geq 2$ such that there is a group of order $m$ with no nontrivial normal $p$ -subgroup for any prime $p$ ?
57. (Jan 17 #4) We call a group $G$ polycyclic if it contains a series of subgroups $\{e\}=G_{0} \subset G_{1} \subset G_{2} \subset \cdots \subset G_{n}=G$ such that $G_{i} / G_{i-1}$ is a (possibly infinite) cyclic group.
(a) Show that a finite group is polycyclic if and only if it is solvable.
(b) Show that $\mathbb{Q}$ is an example of an abelian group which is not polycyclic.
58. (Jan $17 \# 5$ ) Given a finite group $G$ and two subgroups $H, K$ , the double cosets of $H$ and $K$ are the sets of the form $H g K$ for some $g \in G$ .
(a) Show that any two double cosets must be equal or disjoint.
(b) Show that the size of any double coset must divide the product of the orders $|H| \cdot|K|$ .
(c) Find an example of a double coset whose size does not divide the order $|G|$ .
59. (Jan 17 #6)
(a) Find the smallest integer $n$ such that $S_{n}$ has a subgroup of order 10, but $S_{k}$ for $k<n$ does not.
(b) Find the smallest integer $m$ such that $S_{m}$ has an element of order 10 , but $S_{k}$ for $k<m$ does not.
60. (Aug $17 \# 1$ ) Denote by $D_{2 n}$ the dihedral group of order $2 n, n=1$ being admitted. For which natural numbers $m$ and $n$ is $D_{4 n m}$ isomorphic to the direct product $D_{2 m} \times D_{2 n}$ ?
61. (Jan 18 #1) Classify, up to isomorphism, all finite groups of order $2 p$ , where $p$ is a prime number.
62. (Aug 18 #1) Let $G$ be a group, and let $\gamma_{i}(G)$ denote the $i^{\text {th }}$ term of the lower central series of $G$ . That is, $\gamma_{1}(G)=G$ and inductively $\gamma_{i+1}(G)=\left[G, \gamma_{i}(G)\right]$ . Let $\operatorname{IAut}(G) \leq A u t(G)$ be the group of automorphisms of $G$ which induce the trivial automorphism of $G /[G, G]$ . Prove the following statements.
(a) For each $i \geq 1$ , we have $\gamma_{i}(G) \leq G$ is characteristic and $\gamma_{i}(G) / \gamma_{i+1}(G)$ is abelian.
(b) If $\phi \in I A u t(G)$ and $i \geq 1$ , then $\phi$ induces the trivial automorphism of $\gamma_{i}(G) / \gamma_{i+1}(G)$ . You may use without proof the fact that for any group $G$ , there is an inclusion

$\left[[G, G], \gamma_{i}(G)\right] \subseteq \gamma_{i+2}(G)$

(Jan $19 \# 1$ ) Let $X$ and $Y$ be non-abelian simple groups.
(a) Let $G=X \times Y$ . Prove that the only normal subgroups of $G$ are $G, X \times\{1\},\{1\} \times Y$ and the trivial subgroup.
Hint: Show that if $N$ is a normal subgroup of $G$ not contained in $X \times\{1\}$ (respectively, $\{1\} \times Y)$ , then $N$ contains an element of the form $(1, y)$ with $y \neq 1$ (respectively, $(x, 1)$ with $x \neq 1$ ).
(b) Use (a) to prove that $\operatorname{Aut}(X \times X)$ is isomorphic to a semi-direct product of $\operatorname{Aut}(X) \times A u t(X)$ and $\mathbb{Z}_{2}$ (a cyclic group of order 2).
(Jan $19 \# 2$ ) Let $G$ be a finite group of order $n$ .
(a) Prove that there exists an injective homomorphism $\phi: G \rightarrow S_{n}$ such that for every $g \in G$ , the permutation $\phi(g)$ is a product of $n / k$ disjoint cycles of length $k$ (for some $k$ depending on $g$ ).
(b) Now assume that $n$ is even and a Sylow 2 -subgroup of $G$ is cyclic. Use (a) to prove that $G$ has a subgroup of index 2.
(Aug $19 \# 2$ ) Let $n \geq 3$ be an integer and let $S_{n}$ be the symmetric group on $\{1,2, \ldots, n\}$ . Let $H$ be a subgroup of $S_{n}$ with $\left[S_{n}: H\right]=n$ . Prove that

$H \cong S_{n-1}$

Hint: Start by constructing a suitable action of $S_{n}$ associated to $H$ . You may use the description of normal subgroups of $S_{n}$ without proof.
66. (Jan 20 #1)
(a) Let $C$ be a cyclic group of order $n \geq 2$ . Explain briefly why $\operatorname{Aut}(C) \cong \mathbb{Z}_{n}^{\times}$ (the multiplicative group of $\mathbb{Z}_{n}$ ) and why $\operatorname{Aut}(C) \cong \mathbb{Z}_{n-1}$ if $n$ is prime.
(b) Let $G$ be a finite group, let $p$ be the smallest prime dividing $|G|$ , and suppose that $G$ contains a normal subgroup $C$ of order $p$ . Prove that $C$ lies in the center of $G$ .
67. (Aug $20 \# 2$ ) Let $n \geq 3$ be an integer. Denote by $S_{n}$ the symmetric group on $\{1, \ldots, n\}$ and by $D_{2 n}$ the dihedral group of order $2 n$ .
(a) Prove that for every $k \geq 3$ , there exists an injective homomorphism $\phi: D_{2 k} \rightarrow S_{k}$ whose image contains a $k$ -cycle.
(b) Prove (using (a) or otherwise) that every element of $S_{n}$ can be written as a product of two elements of order $\leq 2$ .
68. (Jan 21 #5) Let $n \geq 2$ be an integer, let $g=(1,2,3, \ldots, n) \in S_{n}$ and $H=\langle g\rangle$ .
(a) Prove that the centralizer of $g$ in $S_{n}$ is equal to $H$ .
(b) Now assume that $n$ is prime, and let $N$ be the normalizer of $H$ in $S_{n}$ . Prove that $|N|=n(n-1)$ .
69. (Aug 21 #3)
(a) Let $p$ be a prime and $G$ a group or order $p^{3}$ . Prove that any two elements $x, y$ of $G$ which are conjugate in $G$ commute (that is, $x y=y x$ ). Hint: Prove that any element $g$ of $G$ is contained in some abelian normal subgroup of $G$ (which depends on $g$ ).
(b) Give, with arguments, an example of a group $G$ of order 16 and two elements $x, y$ of $G$ which are conjugate in $G$ but do not commute.
70. (Jan 22 #2) Show that there exist precisely 3 isomorphism classes of groups $G$ that contain a subgroup $H$ of index 2 which is infinite cyclic (i.e., is isomorphic to $\mathbb{Z}$ ). Hint: Consider separately the cases where $G$ is abelian and $G$ is non-abelian. In the non-abelian case consider a natural action of $G$ on $H$ and use it to show that $G$ must have an element of order 2 .

1.1 General Group Theory

Related Problems