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Subgroups and Quotient Groups

Basic Definitions

Generating sets

Definition: Suppose $G$ is a group and $A$ is a subset of $G$ (not necessarily a subgroup, just a bunch of elements). The subgroup $\langle A\rangle$ of $G$ generated by $A$ is \(\langle A\rangle = \bigcap H\) where the intersection is over all subgroups $H$ of $G$ that contain the set $A$.

Some special types of subgroups

See DF Section 2.2

Suppose $G$ is a group and $H$ is a subgroup of $G$.

  1. The centralizer $C_{G}(H)$ of $H$ is the set of elements $g\in G$ such that $gh=hg$ for all $h\in H$.
  2. The normalizer $N_{G}(H)$ of $H$ is the set of elements $g\in G$ such that $gHg^{-1}=H$. In other words, $ghg^{-1}\in H$ for all $h\in H$.
  3. $H$ is a normal subgroup if $N_{G}(H)=G$.
  4. The center $Z(G)$ of $G$ is the the of elements $z$ of $G$ such that $zg=gz$ for all $g\in G$.
  5. If $f:G\to H$ is a homomorphism, the kernel of $f$ is the set of $g\in G$ such that $f(g)=e$.

Notice that:

  1. $C_{G}(H)\subset N_{G}(H)$
  2. The center $Z(G)$ is a normal subgroup of $G$.
  3. $H\subset N_{G}(H)$ and $H$ is a normal subgroup of $N_{G}(H)$.
  4. The kernel of any homomorphism is a normal subgroup. (In fact, the converse is true as well, as we will see later).

Subgroups from group actions

Suppose that $X$ is a set and $G$ acts on $X$. Remember that one way to think of this is that we have a homomorphism from $G$ to $S(X)$. Another way is that we have a map $G\times X\to X$ satisfying $ex=x$ and $g(h(x))=(gh)(x)$ for all $x\in X$ and all $g,h\in G$. Such an action yields subgroups of $G$ as follows:

  1. The kernel of the action is the set of $g\in G$ such that $gx=x$ for all $x\in X$. In other words, the kernel of the action is the kernel of the homomorphism from $G$ to $S(X)$ corresponding to the action. The kernel of the action is therefore a normal subgroup of G.
  2. If $x\in X$, the set of elements $g\in G$ such that $gx=x$ is a subgroup of $G$ called the stabilizer of $x$.

Normalizers and centralizers via group actions

One way to think of the normalizer of $H$ in $G$ as being the largest subgroup of $G$ in which $H$ is normal. Alternatively one can think of them in terms of group actions.

Let $\mathcal{P}(G)$ be the power set of $G$ – that is, the set of subsets of $G$. If $S\subset\mathcal{P}(G)$ is a subset, define \(g(S)=\{gsg^{-1} : s\in S\}.\) This defines an action of $G$ on $\mathcal{P}(S)$. In general the operation $h\mapsto ghg^{-1}$ is called conjugation of $h$ by $g$.

If we choose $S=H$, then by definition $N_{G}(H)$ is exactly the stabilizer of $H$ for this action.

If we restrict the action of $N_{G}(H)$ to the set $H$, then $C_{G}(H)$ is exactly the subset of $N_{G}(H)$ that fixes $H$ pointwise. In other words, $C_{G}(H)$ is the kernel of the conjugation action of $N_{G}(H)$ on $H$.

Cosets

See DF Section 3.1

Definition: Let $H$ be a subgroup of a group $G$. A set $gH=\lbrace gh : h\in H\rbrace$ is called a left coset of $H$. The corresponding set $Hg$ is called a right coset.

Basics

  1. $gH=H$ if and only if $g\in H$.
  2. Any two left (right) cosets are in bijection with each other, so if $H$ is finite all cosets of $H$ have the same number of elements as $H$.
  3. Two left (right) cosets are either equal or disjoint.
  4. There is a bijection between the set of left cosets and the set of right cosets of $H$ in $G$ given by $f(gH)=Hg^{-1}$.
  5. Together, the left (right) cosets of $G$ form a partition of $G$ into disjoint sets.
  6. The index of $H$ in $G$, written $[G:H]$, is the number of left (right) cosets, if that number is finite; otherwise we say $H$ has infinite index in $G$.
  7. $H$ is normal if and only if $gH=Hg$ for every $g\in G$.

One way to obtain most ofthe key properties of cosets is to observe that the relation $x\sim y$ defined by $x=yh$ for some $h\in H$ – or, expressed another way, that $x\in yH$ – is an equivalence relation.

Theorem: (Lagrange) If $G$ is finite and $H$ is a subgroup of $G$ then $\mid H\mid [G:H]=\mid G \mid$.

Corollary: In a finite group, the order of an element divdes the order of the group.

Quotient Group

If $H$ is a normal subgroup, then the set of left (right) cosets of $G$ form a group called the quotient group $G/H$. The group law is $(aH)(bH)=(ab)H$.

The key ingredient of this definition is that the product is well defined. In other words, if $aH=a’H$ and $bH=b’H$ then $(ab)H=(a’b’)H$. Since $H$ is normal, $xH=Hx$ for any $x\in G$. We have $a=a’h$ and $b=b’k$ for elements $h,k$ in $H$. Then $a’hb’k=a’b’h’k$ since $Hb=bH$ by normality of $H$. This shows that $a’b’H=abH$.

Universal property

Let $\pi_{H}:G\to G/H$ be the “canonical map” that sends $g\mapsto gH$.

  • The kernel of this map is $H$.
  • Let $K$ be any group and let $f: G\to K$ be a homomorphism such that $H$ is contained in the kernel of $f$. Then there is a unique homomorphism $\overline{f}:G/H\to K$ such that $f=\overline{f}\pi_{G}$.

The map $\overline{f}$ is defined by $\overline{f}(aH)=f(a)$. This is well defined since $f(h)=e$ for all $h\in H$.

Corollary: A subgroup $H$ is normal if and only if it is the kernel of a homomorphism.

In fact if $H$ is normal then $H$ is the kernel of the homomorphism $\pi_{G}$.