1.3 Abelian Groups

Know basic definitions: Free abelian group, rank, torsion, direct sum, elementary divisors, invariant factors.

Know the Fundamental Theorem of Finitely Generated Abelian Groups:
Every finitely generated abelian group $A$ is of the form $A=T(A) \oplus F(A)$, where $T(A)$ is the (finite) torsion subgroup of $A$ and $F(A)$ is a free abelian group of finite rank $n \in \mathbb{N}_{0}$ (i.e. $F(A)$ is isomorphic to $\mathbb{Z}^{n}$ ), with $n$, the "Betti number" of $A$, uniquely determined by $A$.
Regarding $T(A)$ (which is of course $A$ if $A$ is finite):
Elementary Divisor Form (longest decomposition into cyclics)
$T(A)=\mathbb{Z}_{p_{1}^{f_{1}}} \oplus \cdots \oplus \mathbb{Z}_{p_{s}^{f_{s}}}$ where the $p_{i}$ ’s are (not necessarily distinct) prime numbers, and the number of summands of type $\mathbb{Z}_{p^{f}}$ (the elementary divisors) are invariants of $T(A)$ and $A ; T(A)$ is cyclic iff all elementary divisors have distinct primes.
Invariant Factor Form (shortest decomposition into cyclics)
$T(A)=\mathbb{Z}_{d_{1}} \oplus \cdots \oplus \mathbb{Z}_{d_{l}}$ for $d_{1}\left|d_{2}\right| \cdots \mid d_{l}$ where the invariant factors $d_{i} \in \mathbb{N}$ with $d_{i}>1$ are invariants of $T(A)$ and of $A ; T(A)$ is cyclic iff $l=1$.

Know How to Count: the number of non-isomorphic finite abelian groups of order $n= p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{r}^{e_{r}}$, where this time the $p_{i}$ ’s are distinct primes, is $\mathcal{P}\left(e_{1}\right) \mathcal{P}\left(e_{2}\right) \cdots \mathcal{P}\left(e_{r}\right)$ for $\mathcal{P}(e)=$ number of partitions of the exponent $e$ (the number of ways of writing $e=f_{1}+\cdots+f_{s}$ for $1 \leq f_{1} \leq \cdots \leq f_{s}$ ).

Know How to Decompose:
(I) Write $A=F / K$ for $F$ free on $N$ generators $\left\{x_{1}, \ldots, x_{N}\right\}$ with $M$ relations $B \vec{x}=\overrightarrow{0}$ (ie. $b_{i 1} x_{1}+ \cdots+b_{i N} x_{N}=0$ for $i=1,2, \ldots, M$ ).
(II) Rewrite equations $B \longrightarrow U B$ for elementary $U$ (elementary row operations), change basis of free $\mathbb{Z}$-module $F$ into $\left\{y_{1}, \ldots, y_{N}\right\}$ by $B \longrightarrow B V$ for elementary $V$ (elementary column operations), until reach "diagonal form" $\operatorname{diag}\left(d_{1}, \ldots, d_{n}\right)$ with $n \leq \max \{M, N\}, d_{1}, \ldots, d_{n} \in \mathbb{N}$ and $d_{1}\left|d_{2}\right| \ldots \mid d_{n}$. (III) Then $F=\bigoplus_{i=1}^{N} \mathbb{Z} y_{i}, K=\bigoplus_{i=1}^{n} \mathbb{Z} d_{i} y_{i}$ and $A=F / K \cong \bigoplus_{i=1}^{n} \mathbb{Z} / d_{i} \mathbb{Z} \oplus \mathbb{Z}^{N-n}=\bigoplus_{d_{i} \neq 1} \mathbb{Z}_{d_{i}} \oplus \mathbb{Z}^{N-n}$. So the Betti number of $A$ is $N-n, T(A) \cong \bigoplus_{d_{i} \neq 1} \mathbb{Z}_{d_{i}}$, and the invariant factors of $T(A)$ are $d_{r+1}\left|d_{r+2}\right| \ldots \mid d_{n}$ if $d_{1}=\ldots=d_{r}=1$ and $d_{r+1}>1$.
(Figuratively speaking: reduce the matrix to the diagonal form using elementary row and column operations.)

Know how to go back and forth between Invariant Factors and Elementary DiviSORS: InvFac $\longrightarrow$ ElDiv by factoring each $d_{i}=\prod p_{i j}^{e_{i j}}$, ElDiv $\longrightarrow$ InvFac by $d_{N}=\prod p_{i}^{\text {highest } e_{i}}, d_{N-1}= \prod p_{i}^{\text {next highest } e_{i}}$, etc.

Related Problems

1. Describe (up to isomorphism) all abelian groups of order: [May 91 #4a] 4851; [85 #1c] 2334; [Aug 88 #4] 200; [Jan 87 #1] 72; [Feb 84 #1] 1984; [Sep 80 #1] 80; [May 80 #1] 375.

2. How many non-isomorphic abelian groups are there of order: [Sep 83 #1] 1776, [Jan 98 #2a] 360.

3. (April 77 #3) Use the Fundamental Theorem of Finite Abelian Groups to show the multiplicative group of the ring $\mathbb{Z}_{p^{n}}$ for prime $p, n>1$ has a subgroup of order $p$.

4. (May 78 #6) Give an example of a torsion-free abelian group which is not free. Can you give an example which is finitely generated?

5. (Mar 83 #6) Show a finite abelian group is cyclic iff it has no subgroup isomorphic to $B \oplus B$ for $B \neq 0$.

6. (Aug 89 #6) Find the structure of all abelian groups generated by 3 elements $a, b, c$ satisfying relations $-4 a+2 b+6 c=0,-6 a+2 b+6 c=0,7 a+4 b+15 c=0$.

7. (Jan 92 #1) Show that an infinite abelian group is cyclic iff every nonzero subgroup has finite index.

8. (Aug $94 \# 2$ ) If $G$ is a finite abelian group of order $n$, show that $G$ has a subgroup of order $d$ for each divisor $d$ of $n$. Show that this need not be true if $G$ is not abelian.

9. (Aug 95 #3) Find the order of the abelian group generated by $x, y, z$ subject to the relations $4 x-2 y+4 z=0,7 x-8 y+z=0,8 x+y+13 z=0$.

10. (Aug $96 \# 2$ ) Determine all pairs of positive integers $a, b$ with $a \leq b$ such that $\mathbb{Z}_{a} \times \mathbb{Z}_{b}$ is isomorphic to $\mathbb{Z}_{15} \times \mathbb{Z}_{18} \times \mathbb{Z}_{20}$.

11. (Jan $97 \# 2$ ) Let $G$ be a finite abelian group of order $k$ and $n$ an integer. Show that if $k_{n}$ is the number of solutions of $g^{n}=1$ in $G$, and $k^{(n)}$ is the number of $n$-th powers in $G$, then $k=k_{n} k^{(n)}$.

12. (Jan 98 #2) Take the free abelian group on three generators $x, y, z$, and divide by the relations $2 x+4 y+5 z=0,6 x+8 y+10 z=0,8 x+12 y+20 z=0$. Write the resulting group as a direct sum of cyclic groups.

13. (Aug $98 \# 1 \mathrm{~d}$ ) If $G$ is a finite abelian group of order $p q r$ for distinct primes $p, q, r$, show that $G$ is cyclic.

14. (Aug 98 #3) For an abelian group $A$, the dual group $A^{*}$ is defined to be $\operatorname{Hom}(A, \mathbb{T})$ where the circle group or torus $\mathbb{T}$ is the multiplicative group of complex numbers of modulus 1 (the unit circle in the complex plane). Here $\operatorname{Hom}(A, B)$ denotes the abelian group of homomorphisms of $A$ into $B$ (under $(f+g)(a)=f(a)+g(a)$ ); you may use the additivity property $\operatorname{Hom}\left(A_{1} \oplus\right. \left.A_{2}, B\right) \cong \operatorname{Hom}\left(A_{1}, B\right) \oplus \operatorname{Hom}\left(A_{2}, B\right)$ and the isomorphism property that if $A \cong A^{\prime}, B \cong B^{\prime}$ then $\operatorname{Hom}(A, B) \cong \operatorname{Hom}\left(A^{\prime}, B^{\prime}\right)$.
    (a) Prove that $\mathbb{Z}_{n}^{*}$ is cyclic of order $n$.
    (b) Prove that $A^{*}$ is isomorphic to $A$ for any finite abelian group $A$.

15. (Aug 99 #3) The exponent $e$ of a group $G$ is defined as the smallest positive integer $k$ such that $x^{k}=1$ for all $x \in G$; for an abelian group, in additive notation this is the smallest $k$ such that $k x=0$ for all elements. If $n_{1}, n_{2}, \ldots, n_{r}$ are the invariant factors of a finite abelian group
    $A$ (so $n_{r}\left|n_{r-1}\right| \cdots\left|n_{2}\right| n_{1}$ ), prove that $A$ has exponent $e=n_{1}$, and has an element of order precisely $e$. Conclude that $A$ has an element of order $m$ iff $m$ divides the largest invariant factor $n_{1}$.

16. (Jan $00 \# 3$ )(a) Let $A$ be an abelian group, and let $f$ and $g$ be any two endomorphisms of $A$ (group homomorphisms of $A$ into itself). Let $B:=\operatorname{Fix}(f g)=\{a \in A \mid f(g(a))=a\}$, $C:=\operatorname{Fix}(g f)=\{a \in A \mid g(f((a))=a\}$. Show that $B$ and $C$ are isomorphic subgroups of $A$. (b) How many abelian groups (up to isomorphism) are there of order 1000 which have no elements whose orders are larger than 35 ?

17. (Aug 01 #3) How many nonisomorphic abelian groups are there of order 325 ?

18. (Jan 04 #4) Let $H$ be the subgroup of $G:=\mathbb{Z} \oplus \mathbb{Z}=\{(m, n) \mid m, n \in \mathbb{Z}\}$ generated by $(1,2)$ and $(3,4)$. Identify the quotient group $G / H$.

19. (Aug $04 \# 5$ ) Let $N$ be the $\mathbb{Z}$-submodule of $\mathbb{Z}^{3}$ generated by the column vectors $(2,2,-2)^{t},(-4,-2,4)^{t}$ and $(2,4,4)^{t} \in \mathbb{Z}^{3}$.
    (a) Determine the structure of the abelian group $\mathbb{Z}^{3} / N$.
    (b) Determine a basis $y_{1}, y_{2}, y_{3}$ of $\mathbb{Z}^{3}$ and natural numbers $d_{1}\left|d_{2}\right| d_{3}$ such that $d_{1} y_{1}, d_{2} y_{2}, d_{3} y_{3}$ is a $\mathbb{Z}$-basis of $N$.

20. (Aug $05 \# 8$ ) Let $A$ be a finitely generated abelian group and $n$ a natural number.
    (a) Show that $A$ can be generated by $n$ elements if and only if $A$ is a homomorphic image of $\mathbb{Z}^{n}$.
    (b) If $B$ is a subgroup of $A$, and $A$ can be generated by $n$ elements, prove that also $B$ can be generated by $n$ elements.
    (Hint: It is known from the structure theory of modules over PID’s that this statement is true for $A=\mathbb{Z}^{n}$.)
    (c) If $p$ is a prime number, prove that no subgroup of $\mathbb{Z}^{2} \oplus \mathbb{Z}_{p^{3}} \oplus \mathbb{Z}_{p^{4}}$ is isomorphic to $\mathbb{Z}_{p} \oplus \mathbb{Z}_{p} \oplus \mathbb{Z}_{p}$.

21. (Jan $06 \# 2$ ) Find the number of elements of order precisely $p^{2}$ in the group $\mathbb{Z}_{p^{3}} \times \mathbb{Z}_{p^{5}}$.

22. (Aug 08 #4) Let G be a finite abelian group, $|G|=n$, and define the exponent of $G$ to be $\operatorname{Exp}(G):=\min \left\{k \in \mathbb{N} \mid g^{k}=e\right.$ for all $\left.g \in G.\right\}$.
    a. Prove that $\operatorname{Exp}(G)$ divides $n$, and $n$ divides $\operatorname{Exp}(G)^{j}$ for some $j \in \mathbb{N}$.
    b. If $G, H$ are finite abelian groups with $\operatorname{Exp}(G)=\operatorname{Exp}(H)$ and $|G|=|H|=n$ where $p^{4}$ does not divide $n$ for any prime $p$, prove that $G$ and $H$ are isomorphic.
    c. State (without proof) results analogous to (a) and (b) which hold for $n \times n$ matrices over an algebraically closed field.

23. (Jan $09 \# 2$ ) (a) Classify abelian groups of order $72=2^{3} \cdot 3^{2}$ up to isomorphism (the answer is sufficient).
    (b) Let $m$ and $n$ be positive integers. What is the number of elements in $\mathbb{Z} / n \mathbb{Z}$ whose order divides $m$ ?
    (c) Let $G$ and $H$ be finite abelian groups, and assume that for any $m \in \mathbb{N}$ the groups $G$ and $H$ have the same number of elements of order $m$. Prove that $G$ and $H$ are isomorphic.

24. (Jan $20 \# 2$ ) Let $A$ be the subgroup of $\mathbb{Z}^{2}$ generated by $(2,6)$ and $(4,8)$. Prove that $A \cong \mathbb{Z}^{2}$ and find (with proof) elements $v_{1}, v_{2} \in \mathbb{Z}^{2}$ and $n_{1}, n_{2} \in \mathbb{N}$ such that $\mathbb{Z}^{2}=\mathbb{Z} v_{1} \oplus \mathbb{Z} v_{2}$ and $A=\mathbb{Z}\left(n_{1} v_{1}\right) \oplus \mathbb{Z}\left(n_{2} v_{2}\right)$.

25. (Aug $20 \# 1$ ) Let $A$ be a finitely generated abelian group. Prove that the following are equivalent:
    (a) $\operatorname{dim}_{\mathbb{Q}}\left(A \otimes_{\mathbb{Z}} \mathbb{Q}\right) \leq 1$
    (b) $\operatorname{Aut}(A)$ is finite.

Hint: Use classification of finitely generated abelian groups. What does the condition $\operatorname{dim}_{\mathbb{Q}}\left(A \otimes_{\mathbb{Z}} \mathbb{Q}\right) \leq 1$ tell you about a standard decomposition of $A$ ?
26. (Jan 21 #4) Let $G$ be a finite abelian group and let $p$ be a prime. Prove that the number of elements of order $p$ in $G$ is equal to the number of nontrivial homomorphisms from $G$ to $\mathbb{Z}_{p}$.
Hint: Calculate both numbers separately.
27. (Aug 21 #2) Given a group $G$, denote by $\operatorname{Aut}(G)$ its group of automorphisms.
(a) Let $G=\mathbb{Z}_{p^{a}} \oplus \mathbb{Z}_{p^{b}}$ where $a \geq b>0$. Show that $\operatorname{Aut}(G)$ is non-abelian by explicitly constructing two noncommuting automorphisms. Hint: It may be helpful to start with the case $a=b=1$.
(b) Let $G$ be a finite abelian group. Show that $\operatorname{Aut}(G)$ is abelian $\Longleftrightarrow G$ is cyclic. Hint: Use (a). Remember that you are allowed to do this even if you did not solve (a).

Chapter 2

Rings