3.6: Exercises

[x14] Let \(G\) be the quaternion group ([bd8c]). Prove that \(G\) can’t be written as a semidirect product in any nontrivial fashion.

[x15] Let \(G\) be a group of order \(mn\), where \(m\) and \(n\) have no common factor. If \(G\) contains exactly one subgroup \(M\) of order \(m\) and exactly one subgroup \(N\) of order \(n\), prove that \(G\) is the direct product of \(M\) and \(N\).

[x16] Prove that \(\GL_{2}(\mathbb{F}_{2}\mathbb{)}\approx S_{3}\).

[x17] Let \(G\) be the quaternion group ([bd8c]). Prove that \(\Aut(G)\approx S_{4}\).

[x18] Let \(G\) be the set of all matrices in \(\GL_{3}(\mathbb{R})\) of the form \(\left( \begin{smallmatrix} a & 0 & b\\ 0 & a & c\\ 0 & 0 & d \end{smallmatrix} \right)\), \(ad\neq0\). Check that \(G\) is a subgroup of \(\GL_{3}(\mathbb{R})\), and prove that it is a semidirect product of \(\mathbb{R}^{2}\) (additive group) by \(\mathbb{R}^{\times}\times\mathbb{R}^{\times}\). Is it a direct product of these two groups?

[x19] Find the automorphism groups of \(C_{\infty}\) and \(S_{3}\).

[x19a]Let \(G=N\rtimes Q\), where \(N\) and \(Q\) are finite groups, and let \(g=nq\) be an element of \(G\) with \(n\in N\) and \(q\in Q\). Denote the order of an element \(x\) by \(o(x).\)

(a) Show that \(o(g)=k\cdot o(q)\) for some divisor \(k\) of \(|N|\).

(b) When \(Q\) acts trivially on \(N\), show that \(o(g)=\mathrm{lcm}(o(n),o(q)).\)

(c) Let \(G=S_{5}=A_{5}\rtimes Q\) with \(Q=\langle(1,2)\rangle\). Let \(n=(1,4,3,2,5)\) and let \(q=(1,2)\). Show that \(o(g)=6\), \(o(n)=5\), and \(o(q)=2\).

(d) Suppose that \(G=(C_{p})^{p}\rtimes Q\), where \(Q\) is cyclic of order \(p\) and that, for some generator \(q\) of \(Q\),

\[q(a_{1},\ldots,a_{n})q^{-1}=(a_{n},a_{1},\ldots,a_{n-1}). \nonumber \]

Show inductively that, for \(i\leq p\),

\[\left( (1,0,\ldots,0),q\right) ^{i}=\left( (1,\ldots,1,0,\ldots,0\right) ,q^{i}) \nonumber \]

(\(i\) copies of \(1\)). Deduce that \(\left( (1,0,\ldots,0),q\right)\) has order \(p^{2}\) (hence \(o(g)=o(n)\cdot o(q)\) in this case).

(e) Suppose that \(G=N\rtimes Q\), where \(N\) is commutative, \(Q\) is cyclic of order \(2\), and the generator \(q\) of \(Q\) acts on \(N\) by sending each element to its inverse. Show that \((n,1)\) has order \(2\) no matter what \(n\) is (in particular, \(o(g)\) is independent of \(o(n)\)).

[x19b]Let \(G\) be the semidirect \(G=N\rtimes Q\) of its subgroups \(N\) and \(Q\), and let

\[C_{N}(Q)=\{n\in N\mid nq=qn\text{ for all }q\in Q\} \nonumber \]

(centralizer of \(Q\) in \(N\)). Show that

\[Z(G)=\{n\cdot q\mid n\in C_{N}(Q)\text{, }q\in Z(Q)\text{, }nn^{\prime}% n^{-1}=q^{-1}n^{\prime}q\text{ for all }n^{\prime}\in N\}. \nonumber \]

Let \(\theta\) be the homomorphism \(Q\rightarrow\Aut(N)\) giving the action of \(Q\) on \(N\) (by conjugation). Show that if \(N\) is commutative, then

\[Z(G)=\{n\cdot q\mid n\in C_{N}(Q)\text{, }q\in Z(Q)\cap\Ker(\theta)\}, \nonumber \]

and if \(N\) and \(Q\) are commutative, then

\[Z(G)=\{n\cdot q\mid n\in C_{N}(Q)\text{, }q\in\Ker(\theta)\}. \nonumber \]

[x19c]A homomorphism \(a\colon G\rightarrow H\) of groups is normal if \(a(G)\) is a normal subgroup of \(H\). The cokernel of a normal homomorphism \(a\) is defined to be \(H/a(G)\). Show that, if in the following commutative diagram, the blue sequences are exact and the homomorphisms \(a,b,c\) are normal, then the red sequence exists and is exact:

[x19d]Let \(N\) and \(H\) be subgroups of \(G\), and assume that \(H\) normalizes \(N\), i.e., \(hNh^{-1}\subset N\) for all \(h\in H\). Let \(\theta\) denote the action of \(H\) on \(N\), \(\theta(h)(n)=hnh^{-1}\). Show that

\[(n,h)\mapsto nh\colon N\rtimes_{\theta}H\rightarrow G \nonumber \]

is a homomorphism with image \(NH\).

[x19e]Let \(N\) and \(Q\) be subgroups of a group \(G\). Show that \(G\) is the semidirect product of \(N\) and \(Q\) if and only if there exists a homomorphism \(G\rightarrow Q\) whose restriction to \(Q\) is the identity map and whose kernel is \(N\).