Subgroups

There are lots of subsets of a group $G$ which still have the structure of a group. These objects are defined more precisely next.

Definition 5.8.1   Let $G$ be a group. A subgroup of $G$ is a subset $H$ of $G$ such that $H$, together with the operation $*$ inherited as a subset of $G$, satisfies the group operations (G1)-(G4) (with $G$ replaced by $H$ everywhere).

Notation: If $G$ is a group then we will denote the statement ``$H$ is a subgroup of $G$" by

$$H \ <\ G.$$

Example 5.8.2   For example, if $G=S_4$, and

$$H=\langle (1, 3),(1, 4)\rangle =\{1,(1, 3), (1, 4), (1, 3, 4), (3, 4), (1, 4, 3)\},$$

then $H\ <\ G$.

Theorem 5.8.3 (Lagrange)   Let $H$ be a subgroup of a finite group $G$. Then $\vert H\vert$ divides $\vert G\vert$.

proof: For $x,y\in G$, define $x\sim y$ if $xH=yH$, where

$$xH=\{x*h\ \vert\ h\in H\}.$$

This is an equivalence relation. (The interested reader can easily verify the reflexive, symmetry, and transitivity properties.) Moreover, the equivalence class of $x$ consists of all elements in $G$ of the form $x*h$, for some $h\in H$, i.e., $[x]=xH$. Let $g_1,...,g_m\in G$ denote a complete set of representatives for the equivalence classes of $G$. Because of the cancellation law for groups, $\vert xH\vert=\vert H\vert$ for each $x\in G$. Furthermore, we know that the equivalence classes partition $G$, so

$$G=\cup_{i=1}^m [g_i]=\cup_{i=1}^m g_iH.$$

Comparing cardinalities of both sides, we obtain $\vert G\vert=\vert g_1H\vert+...+\vert g_mH\vert=m\vert H\vert$. This proves the theorem. $\Box$

Definition 5.8.4   If $H$ and $G$ are finite groups and $H\ <\ G$ then the integer $\vert G\vert/\vert H\vert$ is called the index of $H$ in $G$, denoted $[G:H]=\vert G\vert/\vert H\vert$.

Example 5.8.5   A permutation group $G$ generated by elements $g_1, ..., g_n$ belonging to $S_X$ is a subgroup of $S_X$, i.e., $G < S_X$.

Example 5.8.6   Let

$$A_X=\{ g \in S_X\ \vert\ g \ {\rm is\ even}\}.$$

This is a subgroup of $S_n$ called the alternating subgroup of degree $n$ .

Definition 5.8.7   The center of a group $G$ is the subgroup $Z(G)$ of all elements which commute with every element of $G$:

$$Z(G)=\{z\in G\ \vert\ z*g=g*z,\ {\rm for\ all}\ g\in G\}.$$

Of course, the identity element always belongs to $G$. If the identity element is the only element of $Z(G)$ then we say $G$ has trivial center. On the other hand, $G$ is commutative if and only if $G=Z(G)$.

Exercise 5.8.8   For $H = \langle (1,3),(1,4)\rangle$ and $G=S_4$ as above, $G$ is partitioned into four disjoint sets of the form $gH$. Here are two of them: $H(=(1,3)H=(1,4)H=(1,3,4)H=...)$ and $(1,2,3)H(=(1,2)H=(1,2,3,4)H=...$). Find the other two.

Exercise 5.8.9   Let $G$ be a group written multiplicatively and let $H$ be a subset which is closed under multiplication and taking inverses. Show that $H$ is a (sub)group.

Exercise 5.8.10   Show, as a corollary to the previous Theorem 5.8.3, that Theorem 5.7.7(b) is true.

Exercise 5.8.11   Let $G$ be a group. Show $Z(G)$ is a group.

Exercise 5.8.12   Let $G=S_3$. Determine $Z(G)$.

Exercise 5.8.13   Let $H_1$, $H_2$ and $H_3$ be subgroups of a group $G$. (a) Must $H_1 \cup H_2$ be a subgroup of $G$? If so, prove why; if not, give an example. (b) Must $H_1 \cap H_2$ be a subgroup of $G$? Again, prove or disprove. (c) Must $H_1 \cap H_2\cap H_3$ be a subgroup?

Exercise 5.8.14   Let $G=S_5$. Determine $Z(G)$.

Exercise 5.8.15   The subset $xH$, $x\in G$, introduced in the proof of Lagrange's Theorem 5.8.3 is called a ``left coset'' of $H$ in $G$. (These shall be studied in a later section in more detail.) Show that if $xH$ is a subgroup of $G$ then $x\in H$.