91Ó°ÊÓ

Let \(S\) be a subset of a group \(G\). Show that \(S\) is a subgroup if and only if \(a b^{-1} \in S\) whenever \(a, b \in S\).

Show that the intersection of two subgroups is a subgroup.

Let \(X\) be a subset of a group \(G\). A word on \(X\) is an element \(w\) of \(G\) of the form $$ w=x_{1}^{e_{1}} x_{2}^{e_{2}} \cdots x_{n}^{e_{n}} $$ where \(x_{i} \in X\) and \(e_{i}=\pm 1\). Show that the set of all words on \(X\) is a subgroup of \(G\).

Let \([a, b]\) denote the commutator of \(a\) and \(b\). Show that (a) \([a, b]^{-1}=[b, a]\), (b) \([a, a]=e\) for all \(a \in G\), and (c) \(a b=[a, b] b a\). It is interesting to compare these relations with the familiar commutators of operators.

Show that if \(S\) is a subgroup, then \(S^{2} \equiv S S=S\), and \(t S=S\) if and only if \(t \in S\). More generally, \(T S=S\) if and only if \(T \subset S\).

Show that if \(S\) is a subgroup, then \(S a=S b\) if and only if \(b a^{-1} \in S\) and \(a b^{-1} \in S\left(a S=b S\right.\) if and only if \(a^{-1} b \in S\) and \(\left.b^{-1} a \in S\right)\).

Let \(S\) be a subgroup of \(G\). Show that \(a \triangleright b\) defined by \(a b^{-1} \in S\) is an equivalence relation.

Show that \(C_{G}(x)\) is a subgroup of \(G\). Let \(H\) be a subgroup of \(G\) and suppose \(x \in H .\) Show that \(C_{H}(x)\) is a subgroup of \(C_{G}(x)\).

(a) Show that the only element \(a\) in a group with the property \(a^{2}=a\) is the identity. (b) Now use \(e_{G} \star e_{G}=e_{G}\) to show that any homomorphism maps identity to identity. (c) Show that if \(f: G \rightarrow H\) is a homomorphism, then \(f\left(g^{-1}\right)=[f(g)]^{-1}\).

Establish a bijection between the set of right cosets and the set of left cosets of a subgroup. Hint: Define a map that takes \(S t\) to \(t^{-1} S\).