Chapter 5: Direct and Semidirect Products and Abelian Groups

5.1 Direct Products

Proposition 1: If $G_{1},\dots,G_{n}$ are groups, $\lvert G_{1}\times\dots \times G_{n} \rvert=\lvert G_{1} \rvert\dots \lvert G_{n} \rvert$. Note that if $\lvert G_{i} \rvert=\infty$, for any $i$, the direct product's order is similarly infinite.
Proposition 2: Let $G=G_{1}\times\dots \times G_{n}$.

• For each fixed $i$, the set of elements which have the identity of $G_{j}$ in the $j$th position for all $j \neq i$ is a subgroup of $G$ isomorphic to $G_{i}\cong \{ (1,\dots,1,g_{i},1,\dots,1) \mid g_{i}\in G_{i} \}$.
• For each fixed $i$, define $\pi_{i}:G\to G_{i}$ by $\pi_{i}((g_{1},\dots,g_{n}))=g_{i}$. Then $\pi_{i}$ is a surjective homomorphism with $\mathrm{ker\;\pi_{i}}\cong G_{1}\times\dots \times G_{i-1}\times G_{i+1}\times\dots \times G_{n}$.

Definition: The elementary abelian group is $E_{p^{n}}=\underbrace{ Z_{p}\times\dots \times Z_{p} }_{ n }$, with the property that $x^{p}=1,\forall x \in E_{p^{n}}$.

5.2 The Fundamental Theorem of Finitely Generated Abelian Groups

Definition:

• A group $G$ is finitely generated if $\exists A\subseteq G$ with $\lvert A \rvert$ finite such that $G=\langle A \rangle$. (e.g. $\mathbb{Z}$ is finitely generated because $\mathbb{Z}=\langle 1 \rangle$).
• For each $r\in \mathbb{Z}$ with $r\geq 0$, let $\mathbb{Z}^{r}=\underbrace{ \mathbb{Z}\times\dots \times \mathbb{Z} }_{ r }$, where $\mathbb{Z}^{0}=\mathbf{1}$. The group $\mathbb{Z}^{r}$ is called the free abelian group of rank $r$.

Theorem 3: (Chapter Title) Let $G$ be a finitely generated abelian group.

• Then $G\cong \mathbb{Z}^{r}\times Z_{n_{1}}\times\dots \times Z_{n_{s}}$, provided that $r\geq 0$, $n_{j}\geq 2,\forall j$, and $n_{i+1} \mid n_{i}, \forall_{1}\leq i\leq s -1$.
• $G$ only has one possible (unique) representation of this form.
• $r$ is called the free rank or Betti number of $G$, the integers $n_{1},\dots,n_{s}$ are the invariant factors of $G$, and the representation if called the invariant factor decomposition of $G$.

Fact:

• Two finitely generated abelian groups are isomorphic $\iff$ same free rank and same list of invariant factors.
• A finitely generated abelian group is finite only if $r=0$.
• Every prime divisor of $n$ must divide the first invariant factor $n_{1}$.
• The type of $G$ is $(n_{1},\dots,n_{s})$.

Corollary 4: If $n$ is the product of distinct primes, then up to isomorphism the only abelian group of order $n$ is $Z_{n}$.
Fact: Determining all possible isomorphism classes of abelian groups of order $n$ is solved by factoring $n$ and partitioning into invariant factors.
Theorem 5: Let $G$ be an abelian group of order $n>1$ and let the unique factorization of $n$ into distinct prime powers be $n=p_{1}^{\alpha_{1}}\dots p_{k}^{\alpha_{k}}$.

• Then $G\cong A_{1}\times\dots \times A_{k}$, where $\lvert A_{i} \rvert=p_{i}^{\alpha _{i}}$ (these are not alternating groups).
• For each $A\in \{ A_{1},..,A_{k} \}$ with $\lvert A \rvert=p^{\alpha}$, $A\cong Z_{p^{\beta_{1}}}\times\dots \times Z_{p^{\beta_{t}}}$ with $\beta_{1}\geq\dots\geq\beta_{t}\geq 1$ and $\sum_{i=1}^t \beta_{i}=\alpha$.
• $G$ has only one possible (unique) decomposition of the form in (1), and each $A$ similarly has only a unique decomposition of the form in (2). Note also that $Z_{p^{\beta_{1}}}\times\dots \times Z_{p^{\beta_{t}}}$ are the invariant factors of $A_{i}$.
• The integers $p^{\beta_{i}}$ are called the elementary divisors of $G$. The decomposition in (1),(2) is called the elementary divisor decomposition of $G$.

Fact: Determining all the possible invariant factor decompositions of finitely generated abelian groups of order $n$ is equivalent to determining the number of ways to partition each $\alpha$ (for each $p^{\alpha}$ in the factorization of $n$), then independently combining them.
Proposition 6: Let $m,n\in \mathbb{Z}^+$.

• $Z_{m}\times Z_{n}\cong Z_{mn}\iff(m,n)=1$.
• $n=p_{1}^{\alpha_{1}}\dots p_{k}^{\alpha_{k}}\implies Z_{n}\cong Z_{p_{1}^{\alpha_{1}}}\times\dots \times Z_{p_{k}^{\alpha_{k}}}$.

Example: (Invariant Factors/Cyclic Decomposition $\to$ Elementary Divisors): $\lvert G \rvert=1800$, $G\cong Z_{30}\times Z_{30}\times Z_{2}$ $\longrightarrow$ $E=\{ 2,2,2,3,3,5,5 \}$.
Example: (Elementary Divisors $\to$ Invariant Factors): $\lvert G \rvert=1800$, $E=\{ 2,2,2,3,3,25 \}$ $\longrightarrow$ $G\cong Z_{2*3*25}\times Z_{2*3*1}\times Z_{2*1*1}\cong Z_{150}\times Z_{6}\times Z_{2}$.
Example: (Elementary Divisors):

• $Z_{6}\times Z_{15}$ has $E=\{ 2,3,3,5 \}$ $\implies \cong Z_{2}\times Z_{3}\times Z_{3}\times Z_{5}$.
• $Z_{10}\times Z_{9}$ has $E=\{ 2,5,9 \}$ $\implies \cong Z_{2}\times Z_{5}\times Z_{9}$.

Definition:

• If $G$ is a finite abelian group of type $(n_{1},\dots,n_{t})$, then $t$ is called the rank of $G$.
• The exponent of $G$ is the smallest $n\in \mathbb{Z}^+$ such that $x^{n}=1,\forall x \in G$. If $\nexists n$, then the exponent is $\infty$.

5.4 Recognizing Direct Products

Definition: Let $G$ be a group, $x,y\in G$, and $A,B\subseteq G$ with $A,B\neq \emptyset$.

• $[x,y]=x^{-1}y^{-1}xy$ is called the commutator of $x$ and $y$.
• Define $[A,B]=\langle [a,b] \mid a\in A,b\in B \rangle$.
• Define $G'=\langle [x,y] \mid x,y\in G \rangle$ to be the commutator subgroup of $G$.

Proposition 7: Let $G$ be a group, $x,y\in G$, and $H\leq G$.

• $xy=yx[x,y]$. Importantly, $xy=yx \iff[x,y]=1$.
• $H\trianglelefteq G\iff[H,G]<H$.
• $\sigma[x,y]=[\sigma(x),\sigma(y)]$ for any $\sigma \in \mathrm{Aut}(G)$. Also, $G'\mathrm{\;char\;}G$ and $G/G'$ is abelian.
• $G/G'$ is the largest abelian quotient of $G$, i.e. $H\trianglelefteq G$ and $G/H$ is abelian $\iff G'\leq H$.
• If $\varphi:G\to A$ is any homomorphism of $G$ into an abelian group $A$, then $\varphi$ factors through $G'$, i.e. $G'\leq \mathrm{ker\;\varphi}$.

Proposition 8: Let $H,K\leq G$. The number of distinct ways of writing each element of the set $HK$ in the form $hk$ is $\lvert H\cap K \rvert$. If $H\cap K=1$, each element of $HK$ can be written uniquely as $hk$.
Theorem 9: Let $G$ be a group with $H,K\leq G$ satisfying

• $H,K\trianglelefteq G$.
• $H\cap K=1$. Then, $HK\cong H\times K$.

Definition: If $G$ is a group satisfying Theorem 9, $HK$ is called the internal direct product and $H\times K$ is called the external direct product.

5.5 Semidirect Products

Theorem 10: Let $H,K$ be groups and let $\varphi:K\to \mathrm{Aut}(H)$ be a homomorphism. Let $\cdot$ denote the left action of $K$ on $H$ determined by $\varphi$. Let $G$ be the set of $(h,k)$, then define the following multiplication on $G$: $(h_{1},k_{1})(h_{2},k_{2})=(h_{1}\;k_{1}\cdot h_{2},k_{1}k_{2})$. Then,

• $G$ is a group with order $\lvert H \rvert \lvert K \rvert$.
• $H\cong\{(h,1) \mid h\in H\}$ and $K\cong \{ (1,k)\mid k\in K \}$, and the isomorphic copies of $H,K$ in $G$ are subgroups of $G$.

Now, identifying $H$,$K$ with their isomorphic copies in $G$.

• $H\trianglelefteq G$.
• $H\cap K=1$.
• $\forall h\in H,\forall k\in K$, $khk^{-1}=k\cdot h=\varphi(k)(h)$.

Proposition 11: Let $H,K$ be groups and let $\varphi:K\to \mathrm{Aut}(H)$ be a homomorphism. Then the following are equivalent:

• $H\rtimes K\cong H\times K$
• $\varphi$ is the trivial homomorphism
• $K\trianglelefteq H\rtimes K$.

Theorem 12: Suppose $G$ is a group with $H,K\leq G$ such that

• $H\trianglelefteq G$
• $H\cap K=1$. Let $\varphi:K\to \mathrm{Aut}(H)$ be the homomorphism that maps $k$ to an inner automorphism. Then $HK\cong H\rtimes K$.

Definition: Let $H\leq G$. The complement for $H$ in $G$ is some $K\leq G$ with $G=HK$ and $H\cap K=1$.