1.2 Applications of the Sylow Theorems

Know all parts to the Sylow Theorem: If $|G|=p^{e} m$ for a prime $p$ and a natural number $m$ with $(p, m)=1$, a Sylow $p$-subgroup of $G$ is any subgroup $P$ with $|P|=p^{e}$, i.e. a $p$-subgroup of the maximal possible order.
(i) (Existence) If $p$ is a prime number and $p^{\alpha}$ divides the order of $G$ then $G$ has a subgroup of order $p^{\alpha}$ (in particular, Sylow $p$-subgroups exist).
(ii) (Inclusion) Any $p$-subgroup of a finite group $G$ is contained in a Sylow $p$-subgroup of $G$ (so the Sylow $p$-subgroups of $G$ can also be characterized as the maximal $p$-subgroups of $G$ ).
(iii) (Conjugacy) Any two Sylow $p$-subgroups of $G$ are conjugate.
(iv) (Number) The number $n_{p}$ of them is congruent to $1 \bmod p$, and divides $m$ (the number is exactly the index of the normalizer of any particular $P, n_{p}=\left[G: N_{G}(P)\right]$ ).

Basic fact: Any conjugate of a Sylow subgroup is Sylow, so if for some $p$ there is only one Sylow $p$-subgroup, it must be a normal subgroup (proper unless $G$ itself is a $p$-group, $|G|=p^{e}$, in which case it is nilpotent, hence solvable, hence not simple, unless $|G|=p$ ).

Know the 3 principal methods of proving that a group is NOT SIMPLE (which by induction can often be improved to showing that the group is SOLVABLE):

1. Simple Sylow Count: Use (iv) to show that the number of Sylow $p$-subgroups is 1 . Examples: $|G|=1776$ [Sep $83 \# 1$ ], $1989[$ Jan $89 \# 1], 1995[$ Jan $95 \# 1$; show $G$ solvable], 1001 [Aug $95 \# 2$; show $G$ abelian, cyclic], 200 [Aug $88 \# 4 ; 1985 \# 1$ a]; $n=20,28,44,52 ; n=p^{e} m$ for $p>m$.

2. Small Index: Show that $|G|$ does not divide $n_{p}$ ! for some prime $p$, eg $p^{e}$ doesn’t divide ( $m-1$ )! (Reason: $G$ acts transitively on the set $\mathcal{P}$ of Sylow $p$-subgroups by conjugation, so by simplicity either (i) $G$ embeds faithfully in $\operatorname{Sym}(\mathcal{P})$ of size $n_{p}!$, so $|G|\left|n_{p}!\right| m$ !, or (ii) $G$ is mapped onto the identity, in which case by transitivity there would be only one $P$, contrary to the Basic Fact). Examples: $|G|=24,36,48 ; 72$ [Jan $87 \# 1$, Sep $84 \# 1$ ].

3. Element Count: Use (iv) to get estimates on $n_{p}>1$ for the relevant $p$ ’s, and count how many elements there are of order $p^{k}$ for the various $k$ and $p$ (crucial: two subgroups of prime order $p$ can overlap only in one element; the answer is not so simple for Sylow subgroups of higher order $p^{k}$ ), and show the number of elements would be $>|G|$. Examples: $|G|=30,56$.

Know the following characterization of finite nilpotent groups:

$G$ is nilpotent iff all its Sylow subgroups are normal iff $G$ is the direct product of its Sylow subgroups.

Related Problems

1. (Nov 77 #7) Must a group of order 70 be abelian? Solvable? Can you say anything about its normal subgroups?

2. (a) (January $81 \# 5$, Sep 78 #3) If $G$ of order 60 has exactly 4 elements of order 5 , there is a proper normal subgroup.
   (b)If $G$ of order 60 has more than 4 elements of order 5 , then $G$ is simple (and hence isomorphic to $A_{5}$ ).

3. (January 82 #VIIa, May 89 #8, Sep 93 #3) If a normal subgroup $H$ of $G$ contains a Sylow subgroup $P$ of $G$, then $G=H N_{G}(P)$.

4. (Jan 82 #VIIb,c) The Frattini subgroup $F$ of $G$ is defined to be the intersection of all maximal subgroups of $G$. Show
   (a) $F$ is normal in $G$,
   (b) if $G$ is finitely generated then $F$ is inessential ( $G=F H \Longrightarrow G=H$ ),
   (c) if $F$ contains a $p$-Sylow subgroup of $G$ then that subgroup is normal.

Comments: Do (b) first for finite groups. Then prove for finitely generated $G$ that every proper subgroup of $G$ is contained in a maximal subgroup (mimic the standard proof for
commutative rings with 1 that proper ideals are contained in maximal ideals). In (c) we assume that $G$ is finite.
5. A member $g$ of a group $G$ is called a nongenerator of $G$ if whenever $G$ is generated by a subset containing $g$, it is also generated by the subset with $g$ removed. It is a fact that the set of all nongenerators of $G$ forms a subgroup, the Frattini subgroup of $G$. If $F$ is the Frattini subgroup of a finite $p$-group $G$ ( $p$ a prime), show that $g \in F$ iff $g$ is in every subgroup of index $p$ of $G$. (You may use the fact that if $H$ is a proper subgroup of $G$, then $H$ is a proper subgroup of its normalizer $N_{G}(H)$ ). Conclude that $G / F$ is an abelian group of exponent $p$.
6. (Sep 82 #6) If $G$ is nonabelian of order 21, show it is generated by elements $s, t$ with $s^{7}= t^{3}=1, t^{-1} s t=s^{2}$.
7. (Sep 86 #2) Find ALL groups of order 99.
8. (Fall $87 \# 1$ ) If $G$ of order 160 has two distinct subgroups of order 80 , then $G$ has a normal subgroup of order 5 .
9. (May 90 #2) A group of order 441 is solvable.
10. (May $91 \# 4 \mathrm{~b}$ ) If $G$ is a group of order 231, show the 11-Sylow subgroup $H$ of $G$ is normal in $G$ and lies in the center of $G$. (Hint: Let $G$ act on $H$ by conjugation.)
11. (Jan 92 #4) If a finite group $G$ has a normal $p$-Sylow subgroup $P$, then $\phi(P) \subseteq P$ for every endomorphism $\phi$ of $G$.
12. (Jan 94 #4) Let $f: G \longrightarrow H$ be a surjective homomorphism of finite groups, and let $p$ be a prime.
(a) Prove that if $P$ is a $p$-Sylow subgroup of $G$, then $f(P)$ is a $p$-Sylow subgroup of $H$.
(b) Prove that every $p$-Sylow subgroup of $H$ has the form $f(P)$ for some $p$-Sylow subgroup $P$ of $G$.
13. (Aug $94 \# 1$ ) Let $P$ be a $p$-Sylow subgroup of a finite group $G, p$ a prime dividing the order of $G$.
(a) Prove that $P$ consists of all the $p$-torsion elements of the normalizer $N_{G}(P)$, that is, all elements of $N_{G}(P)$ whose order is a power of $p$. (Hint: apply Sylow to $N_{G}(P)$.)
(b) Prove that $P$ is a characteristic subgroup of $N_{G}(P)$, that is, is invariant under all automorphisms of $N_{G}(P)$.
(c) Prove that $N_{G}\left(N_{G}(P)\right)=N_{G}(P)$.
14. (Aug 96 #8) Give an example of two finite groups whose Sylow subgroups are isomorphic for each prime, but which are not themselves isomorphic.
15. (Jan $97 \# 6$ ) Let $P$ be a $p$-Sylow subgroup of a finite group $G$. Prove that if $H$ is a subgroup of $G$, then for some $g \in G, H \cap g P g^{-1}$ is a $p$-Sylow subgroup of $H$.
16. (Jan 98 #1a) What can you say about groups of the following orders $n$ ? (Give reasons, making free use of any theorems you know.)
(a) $n=2^{4}+1$.
(b) $n=2^{3}+1$.
17. (Aug $98 \# 1 \mathrm{ab}$ ) Let $G$ be a finite group of order $3 \cdot 5 \cdot 17$. Show that the Sylow 17-subgroup is normal. If there exists an element of order 15 in G, show that the Sylow 3- and 5-subgroups are also normal. (Hint: show they are properly contained in their normalizers.)
18. If $G$ and its normal subgroup $N$ have the same power of $p$, then all $p$-Sylow subgroups of $G$ live in $N$; if one is normal in $N$, then it is the unique $p$-Sylow subgroup of $G$.
19. (August $98 \# 1 \mathrm{c}$ ) If all Sylow subgroups of a finite group $G$ are normal and abelian, show that $G$ itself is abelian.
20. (Aug $99 \# 1$ ) Show that if $G \supseteq M \supseteq N_{G}(P)$ for a $p$-Sylow subgroup $P$ of a finite group G , then $[G: M] \equiv 1(\bmod p)$.
21. (Jan $00 \# 5$ ) Let $G$ be a finite group such that every element commutes with its conjugates (for any $g, h \in G$ the elements $h$ and $g h g^{-1}$ commute).
(a) Show that any Sylow subgroup of such a $G$ is normal. [Hard!]
(b) Explain why the group of quaternions $\{ \pm 1, \pm i, \pm j, \pm k\}$ is such a group $G$.
22. (Aug $01 \# 2$ ) Show that if $p$ is a prime and $H$ is a subgroup of $G$, then the number of distinct $p$-Sylow subgroups of $H$ is less than or equal to the number of distinct $p$-Sylow subgroups of $G$.
23. (Aug $03 \# 7$ ) Let $G$ be a finite group of order $n$. Suppose that for every $d$ dividing $n$, the equation $x^{d}=1$ has at most $d$ solutions in $G$. Show that:
(a) For each prime p, the Sylow $p$-subgroup of $G$ is unique, and thus is normal.
(b) The Sylow $p$-subgroup of $G$ is cyclic.
(c) Use (a) and (b) to show that $G$ is cyclic.
24. (Aug 04 #2) Let $G$ be a group of order 56, and let $P_{p}$ for $p \in\{2,7\}$ be a Sylow $p$-subgroup of $G$.
(a) Show that $P_{2}$ or $P_{7}$ is normal in $G$.
(b) Give an example of a group $G$ with $|G|=56$ where $P_{2}$ is not normal.
(c) Show that there exists a group $G$ of order 56 with a non-normal $P_{7}$. You can either do this by exhibiting a concrete example with this property or by describing how to construct such a group. In the latter case you have to justify why your approach works but you needn’t give all details of the construction.
25. (Jan 05 #2) Consider the following two statements:
(a) Any group of order 455 is abelian.
(b) Any group of order 455 is solvable but there exist non-abelian groups of order 455 . Decide which of the two statements is true ( 2 points) and prove it.
26. (Aug $05 \# 1$ ) Let $G$ be a group of order 60 which acts transitively on a set $S$ with 20 elements. Prove that there exist two elements $x, y \in S$ with $x \neq y$ and equal stabilizers $G_{x}=G_{y}$.
27. (Jan $06 \# 3$ ) If a prime $p$ divides the order of a finite nonabelian simple group $G$, show that $|G|<n_{p}$ ! where $n_{p}$ is the number of distinct $p$-Sylow subgroups of $G$.
28. (Aug 06 #1) Prove that there is no simple group of order $2 \times 3^{3} \times 5^{2}$.
29. (Aug $06 \# 3$ ) Show that any nilpotent group $G$ of order 900 is abelian.
30. (Aug 08 #6) Let $G$ be a group of order $p q r$ where $p, q, r$ are prime numbers with $(i) p>q> r,(i i) \operatorname{gcd}(q, p-1)=\operatorname{gcd}(r, q-1)=\operatorname{gcd}(r, p-1)=1$.
a. Prove that $G$ has a normal subgroup $P$ of order $p$.
b. Prove that $G$ has a subgroup $Q$ of order $q$ which commutes with $P$, so that $P \times Q$ is a subgroup of $G$ of order $p q$.
c. Prove that $Q$ is normal, so $P \times Q$ is a normal subgroup of $G$.
d. Prove that $G$ has a subgroup $R$ of order $r$ which commutes with $P \times Q$, so that $P \times Q \times R$ is a subgroup of $G$. Conclude that $G$ is cyclic.
31. (Jan $09 \# 1$ ) Let $G$ be a group of order $660=11 \cdot 60$, and let $P$ be a Sylow 11-subgroup of $G$. Assume that $C_{G}(P)=P$.
(a) Prove that $\left|N_{G}(P)\right|=55$.
(b) Let $H$ be a normal subgroup of $G$. Prove that either $P \subseteq H$ or $|H| \equiv 1 \bmod 11$. Hint: Consider the conjugation action of $P$ on $H$.
32. (Aug 09 #1) Let $G$ be a group of order 56 which does NOT have a normal subgroup of order 8.
(a) Prove that $G$ has a normal subgroup of order 7 .
(b) Prove that $G$ has a subgroup of order 14.
(c) Prove that $G$ has a normal subgroup of order 14.

Remark: Of course, you may omit (b) if you correctly answered (c).
33. (Aug $10 \# 1$ ) Let $p$ be prime. Let $G$ be a finite group, $K$ a normal subgroup of $G$, and assume that $|G / K|$ is divisible by $p$. Let $P$ be a Sylow $p$-subgroup of $G$.
(a) Prove that $P K / K$ is a Sylow $p$-subgroup of $G / K$.
(b) Prove that $n_{p}(G / K)$ divides $n_{p}(G)$ where $n_{p}(\cdot)$ denotes the number of Sylow $p$-subgroups.
(c) prove that $n_{p}(G / K)=n_{p}(G)$ if and only if $P$ is normal in $P K$.
34. (Aug 11 # 7) (a) Let $G$ be a finite simple group of order 168. How many elements of order 7 does G have? Why?
(b) How many conjugacy classes of elements of order 7 does G have? Hint: By looking at Sylow 3-subgroups, show that $G$ has no cyclic subgroup of order 21. Use this to determine the centralizer in G of an element of order 7. . . . .
(c) Assume that you know that $G:=G L_{3}\left(\mathbb{F}_{2}\right)$ is a simple group. Explicitly exhibit two elements of $G$ of order 7 which are not conjugate in $G$. Explain.
35. (Jan $12 \# 1$ ) Let $F=\mathbb{F}_{q}$ be a finite field, where $q=p^{r}$ is a power of a prime $p$. Let $G=G L_{n}(F)$ be the group of all $n \times n$ invertible matrices with entries in $F$. Once you pick an ordered basis of $V:=F^{n}$, you may find it useful to identify $G$ with the group of invertible linear operators on $V$.
(a) Calculate the order of $G$. Explain your answer carefully and write it in the simplest form
as you can.
(b) Determine the order of a Sylow $p$-subgroup of $G$, and explicitly exhibit a Sylow $p$-subgroup $U$ of $G$.
(c) What is the normalizer in $G$ of the Sylow $p$-subgroup $U$ that you exhibited in (b)? An answer is sufficient.
(d) How many Sylow $p$-subgroups of $G$ are there? Explain how your answer in (d) is consistent with Sylow’s theorem.
36. (Aug 12 # 2) Let $G$ be a finite group and $p$ a prime divisor of $|G|$. Assume that every element of $G$ of $p$-power order is contained in a normal $p$-subgroup of $G$. Show that $G$ has only one Sylow $p$-subgroup.
37. (Aug 16 #4) Fix a group $G$.
(a) Show that if $N$ is a normal Sylow $p$-subgroup of $G$ and $H$ a subgroup of order not divisible by $p$, then $H N$ is a subgroup of $G$ isomorphic to a semi-direct product $N \rtimes H$.
(b) Consider a group $G$ of order 255. Show that $G$ is cyclic.
38. (Aug $17 \# 2$ ) Let $G$ be a group of order $16 \cdot 11 \cdot 13 \cdot 17$. Assume that $G$ has a normal nonabelian Sylow 2-subgroup. Show that the center of $G$ is nontrivial.
Remark: The claim remains true if $G$ has an abelian normal Sylow 2-subgroup but then it is a bit harder to prove.
39. (Aug $18 \# 2$ ) Let $G$ be a non-cyclic group of order 57 . Determine the number of elements of all orders of $G$.
40. (Aug 19 #1)
(a) Let $G$ be a group of order 12. Prove that $G$ has a normal Sylow 2-subgroup or a normal Sylow 3-subgroup.
(b) Prove that there are at least 5 pairwise non-isomorphic groups of order 12 (in fact, 5 is the exact number of isomorphism classes, but you are not asked to prove this). Partial credit for exhibiting fewer than 5 non-isomorphic groups (with proof) will be given.
41. (Jan $22 \# 1$ ) Let $p$ be a prime and $G=G L_{2}\left(\mathbb{F}_{p}\right)$, the group of invertible $2 \times 2$ matrices over $\mathbb{F}_{p}$.

(1) Find the order of $G$ (with proof).

(2) Show that $U=\left\{\left(\begin{array}{cc}1 & a \\ 0 & 1\end{array}\right): a \in \mathbb{F}_{p}\right\}$ is a Sylow $p$-subgroup of $G$

(3) Find the normalizer $N=N_{G}(U)$ and the number of Sylow $p$-subgroups of $G$ (with proof).

Hint: In (3) one can solve either part of the problem first and then use the answer to solve the other part.