Chapter 3: Quotient Groups and Homomorphisms

3.1 Quotient Groups

Proposition 1: Let $G,H$ be groups and let $\varphi:G\to H$ be a homomorphism. Then,

Theorem 3: Let $G,H$ be groups and $K=\mathrm{ker\;}\varphi$ with $\varphi:G\to H$. Then the set of left cosets of $K$ in $G$ is $G/K$ and the operation $uK\circ vK=(uv)K$ is well defined if the converse of injectivity is true, i.e. $u=v\implies \varphi(u)=\varphi(v)$.
Proposition 4: Let $G$ be a group and $N\leq G$. Then the set of left cosets of $N$ in $G$ partition $G$, and $\forall u,v\in G$, $uN=vN\iff v^{-1}u\in N$.
Proposition 5: Let $G$ be a group and $N\leq G$. The operation $uN\cdot vN=(uv)N$ is well defined $\iff gng^{-1}\in N,\forall g\in G$ and $\forall n\in N$. If the operation if well defined, the set of left cosets of $N$ in $G$ is a group.
Definition: $gng^{-1}$ is the conjugate of $n\in N$ by $g\in G$. The set $gNg^{-1}=\{ gng^{-1} \mid n\in N \}$ is called the conjugate of $N$ by $g$. The element $g$ normalizes $N$ if $gNg^{-1}=N$. A subgroup $N$ of $G$ is normal ($N\trianglelefteq G$) if $gNg^{-1}=N,\forall g\in G$. All subgroups of an abelian group are normal.
Theorem 6: Let $N\leq G$. The following statements are equivalent:

Proposition 7: $N\trianglelefteq G\iff \exists \varphi$ such that $\varphi$ is a homomorphism and $\mathrm{ker\;\varphi}=N$.
Definition: Let $N\trianglelefteq G$. The homomorphism $\pi:G\to G/N$ defined by $\pi(g)=gN$ is a natural projection/homomorphism. If $\overline{H}\leq G/N$, the complete preimage of $\overline{H}$ in $G$ is the preimage of $\overline{H}$ under the natural projection
Fact: The quotient groups of of a cyclic group are cyclic with order $\frac{\lvert G \rvert}{\lvert N \rvert}$.

3.2 Lagrange's Theorem

Theorem 8: (Lagrange) If $G$ is a finite group and $H\leq G$, then $\lvert H \rvert \Big| \lvert G \rvert$ and $\lvert G:H \rvert=\frac{\lvert G \rvert}{\lvert H \rvert}$. This also implies $\frac{\lvert G:H \rvert}{\lvert H:K \rvert}=\lvert G:K \rvert$.
Corollary 9: If $G$ is a finite group and $x \in G$, $\lvert x \rvert = \lvert \langle x \rangle \rvert \Big| \lvert G \rvert$ and $x^{\lvert G \rvert} =1$.
Corollary 10: If $G$ is a group of prime order $p$, then $G$ is cyclic $\implies G\cong Z_{p}$.
Theorem 11: (Cauchy) If $G$ is a finite group and $p \Big| \lvert G \rvert$, then $\exists x \in G$ such that $\lvert x \rvert=p$.
Theorem 12: (Sylow) If $G$ is a finite group of order $p^{\alpha}m$, where $p$ is a prime and $p \nmid m$, then $\exists H\leq G$ such that $\lvert H \rvert=p^{\alpha}$.
Definition: Let $H,K\leq G$. Then $HK=\{ hk \mid h\in H, k\in K \}$.
Proposition 13: If $H,K\leq G$ and are finite, then $\lvert HK \rvert=\frac{\lvert H \rvert\lvert K \rvert}{\lvert H\cap K \rvert}$.
Proposition 14: If $H,K\leq G$, $HK\leq G\iff HK=KH$.
Corollary 15: If $H,K\leq G$ and $H\leq N_{G}(K)$, then $HK\leq G$. In particular, if $K\trianglelefteq G$ then $HK\leq G$ for any $H\leq G$.

3.3 Isomorphism Theorems

Theorem 16: (1) If $\varphi:G\to H$ is a homomorphism of groups, then $\mathrm{ker\;}\varphi \trianglelefteq G$ and $G/\mathrm{ker\;}\varphi \cong \varphi(G)$.
Corollary 17: Let $\varphi:G\to H$ be a homomorphism of groups. $\varphi$ is injective $\iff \mathrm{ker\;\varphi}=1$, and $\lvert G:\mathrm{ker\;\varphi} \rvert=\lvert \varphi(G) \rvert$.
Theorem 18: (2) Let $G$ be a group, let $A,B\leq G$ and assume $A\leq N_{G}(B)$. Then $AB$ is a subgroup of $G$, $B\trianglelefteq AB$, $A\cap B\trianglelefteq A$ and $AB/B \cong A/A\cap B$.
Theorem 19: (3) Let $G$ be a group and let $H,K\trianglelefteq G$ with $H\leq K$. Then $K/H\trianglelefteq G/H$ and $(G/H)/(K/H)\cong G/K$.
Theorem 20: (4) Let $G$ be a group and let $N\trianglelefteq G$. Then there is a bijection from the set of subgroups $A$ of $G$ which contain $N$ onto the set of subgroups $\overline{A}=A/N$ of $G/N$. In particular, every subgroup of $\overline{G}$ is of the form $A/N$ for some subgroup $A$ of $G$ containing $N$ (namely, its preimage in $G$ under the natural projection homomorphism from $G$ to $G/N$). $\forall A,B\leq G$ with $N\leq A$ and $N\leq B$, this bijection has the following properties:

3.4 Composition Series and the Hölder Program

Proposition 21: If $G$ is a finite abelian group and $p$ is a prime dividing $\lvert G \rvert$, then $\exists x \in G$ with $\lvert x \rvert=p$.
Definition: A (finite or infinite) group $G$ is simple if $\lvert G \rvert > 1$ and the only normal subgroups of $G$ are $1$ and $G$.
Definition: In a group $G$ a sequence of subgroups $1=N_{0}\leq N_{1}\leq\dots\leq N_{k-1}\leq N_{k}=G$ is called a composition series if $N_{i}\trianglelefteq N_{i+1}$ and $N_{i+1}/N_{i}$ is a simple group for $0\leq i\leq k-1$. If the above sequence is a composition series, the quotient groups $N_{i+1}/N_{i}$ are called composition factors of $G$.
Theorem 22: (Jordan-Hölder) Let $G$ be a finite group with $G \neq 1$. Then

• $G$ has a composition series
• The composition factors in a composition series are unique, namely, if $1=N_{0}\leq N_{1}\leq\dots\leq N_{r}=G$ and $1=M_{0}\leq M_{1}\leq\dots\leq M_{s}=G$ are two composition series for $G$, then $r=s$ and there is some permutation $\pi$ of $\{ 1,2,\dots,r \}$ such that $M_{\pi(i)}/M_{\pi(i)-1}\cong N_{i}/N_{i-1}$, for $1\leq i\leq r$.

Theorem: There is a list consisting of 18 (infinite) families of simple groups and 26 simple groups not belonging to these families (the sporadic simple groups) such that every finite simple group is isomorphic to one of the groups in this list.
Theorem: (Feit-Thomopson) If $G$ is a simple group of odd order, then $G\cong Z_{p}$ for some prime $p$.
Definition: A group $G$ is solvable if there exists a chain of subgroups $1=G_{0}\trianglelefteq G_{1}\trianglelefteq \dots\trianglelefteq G_{s}=G$ such that $G_{i+1}/G_{i}$ is abelian for $i=0,1,\dots,s-1$.
Theorem: The finite group $G$ is solvable $\iff$ for every divisor $n$ of $\lvert G \rvert$ such that $\left( n, \frac{\lvert G \rvert}{n} \right)=1$, $G$ has a subgroup of order $n$.

3.5 Transpositions and the Alternating Group

Definition: A $2$-cycle is called a transposition.
Definition:

• $\epsilon(\sigma)$ is called the sign of $\sigma$.
• $\sigma$ is called an even permutation if $\epsilon(\sigma)=1$ and an odd permutation if $\epsilon(\sigma)=-1$.

Proposition 23: The map $\epsilon:S_{n}\to \{ \pm1 \}$ is a homomorphism (where $\{ \pm1 \}\cong Z_{2}$)
Proposition 24: Transpositions are all odd permutations and $\epsilon$ is surjective.
Definition: The alternating group of degree $n$, denoted by $A_{n}$, is the kernel of the homomorphism $\epsilon$ (i.e. the set of even permutations).
Proposition 25: The permutation $\sigma$ is odd $\iff$ the number of cycles of even length in its cycle decomposition is odd.