4.5: Exercises

[x20a] Let \(H_{1}\) and \(H_{2}\) be subgroups of a group \(G\). Show that the maps of \(G\)-sets \(G/H_{1}\rightarrow G/H_{2}\) are in natural one-to-one correspondence with the elements \(gH_{2}\) of \(G/H_{2}\) such that \(H_{1}\subset gH_{2}g^{-1}\).

[x20] (a) Show that a finite group \(G\) can’t be equal to the union of the conjugates of a proper subgroup \(H\).

(b) Show that (a) holds for an infinite group \(G\) provided that \((G\colon H)\) is finite.

(c) Give an example to show that (a) fails in general for infinite groups.

(d) Give an example of a proper subset \(S\) of a finite group \(G\) such that \(G=\bigcup_{g\in G}gSg^{-1}\).

[x20b]Show that any set of representatives for the conjugacy classes in a finite group generates the group.

[x21] Prove that any noncommutative group of order \(p^{3}\), \(p\) an odd prime, is isomorphic to one of the two groups constructed in ([it16e], [it16d]).

[x22] Let \(p\) be the smallest prime dividing \((G:1)\) (assumed finite). Show that any subgroup of \(G\) of index \(p\) is normal.

[x23] Show that a group of order \(2m\), \(m\) odd, contains a subgroup of index \(2\). (Hint: Use Cayley’s theorem [bd11])

[x23a]For \(n\geq5\), show that the \(k\)-cycles in \(S_{n}\) generate \(S_{n}\) or \(A_{n}\) according as \(k\) is even or odd.

[x24] Let \(G=\GL_{3}(\mathbb{F}_{2})\).

1. Show that \((G:1)=168\).

2. Let \(X\) be the set of lines through the origin in \(\mathbb{F}_{2}^{3}\); show that \(X\) has \(7\) elements, and that there is a natural injective homomorphism \(G\hookrightarrow\Sym(X)=S_{7}\).

3. Use Jordan canonical forms to show that \(G\) has six conjugacy classes, with \(1\), \(21\), \(42\), \(56\), \(24\), and \(24\) elements respectively. [Note that if \(M\) is a free \(\mathbb{F}_{2}[\alpha]\)-module of rank one, then \(\End_{\mathbb{F}_{2}[\alpha]}(M)=\mathbb{F}_{2}[\alpha]\).]

4. Deduce that \(G\) is simple.

[x25] Let \(G\) be a group. If \(\Aut(G)\) is cyclic, prove that \(G\) is commutative; if further, \(G\) is finite, prove that \(G\) is cyclic.

[x26] Show that \(S_{n}\) is generated by \((1\,2),(1\,3),\ldots,(1\,n)\); also by \((1\,2),(2\,3),\ldots,(n-1\,n)\).

[x27] Let \(K\) be a conjugacy class of a finite group \(G\) contained in a normal subgroup \(H\) of \(G\). Prove that \(K\) is a union of \(k\) conjugacy classes of equal size in \(H\), where \(k=(G:H\cdot C_{G}(x))\) for any \(x\in K\).

[x28] (a) Let \(\sigma\in A_{n}\). From Exercise [x27] we know that the conjugacy class of \(\sigma\) in \(S_{n}\) either remains a single conjugacy class in \(A_{n}\) or breaks up as a union of two classes of equal size. Show that the second case occurs \(\iff\) \(\sigma\) does not commute with an odd permutation \(\iff\) the partition of \(n\) defined by \(\sigma\) consists of distinct odd integers.

(b) For each conjugacy class \(K\) in \(A_{7}\), give a member of \(K\), and determine \(|K|\).

[x29] Let \(G\) be the group with generators \(a,b\) and relations \(a^{4}=1=b^{2}\), \(aba=bab\).

1. Use the Todd-Coxeter algorithm (with \(H=1\)) to find the image of \(G\) under the homomorphism \(G\rightarrow S_{n}\), \(n=(G:1)\), given by Cayley’s Theorem 1.11. [No need to include every step; just an outline will do.]

2. Use Sage/GAP to check your answer.

[x30] Show that if the action of \(G\) on \(X\) is primitive and effective, then the action of any normal subgroup \(H\neq1\) of \(G\) is transitive.

[x31] (a) Check that \(A_{4}\) has \(8\) elements of order \(3\), and \(3\) elements of order \(2\). Hence it has no element of order \(6\).

(b) Prove that \(A_{4}\) has no subgroup of order \(6\) (cf. [bd18p] ). (Use [ga20].)

(c) Prove that \(A_{4}\) is the only subgroup of \(S_{4}\) of order \(12\).

[x32] Let \(G\) be a group with a subgroup of index \(r\). Prove:

1. If \(G\) is simple, then \((G:1)\) divides \(r!\).

2. If \(r=2,3,\) or \(4\), then \(G\) can’t be simple (except for te trivial cases \(C_{2}\), \(C_{3}\)).

3. There exists a nonabelian simple group with a subgroup of index \(5\).

[x33] Prove that \(S_{n}\) is isomorphic to a subgroup of \(A_{n+2}\).

[x33b]Let \(H\) and \(K\) be subgroups of a group \(G\). A double coset of \(H\) and \(K\) in \(G\) is a set of the form

\[HaK=\{hak\mid h\in H\text{, }k\in K\} \nonumber \]

for some \(a\in G\).

1. Show that the double cosets of \(H\) and \(K\) in \(G\) partition \(G\).

2. Let \(H\cap aKa^{-1}\) act on \(H\times K\) by \(b(h,k)=(hb,a^{-1}b^{-1}ak)\). Show that the orbits for this action are exactly the fibres of the map \((h,k)\mapsto hak\colon H\times K\rightarrow HaK\).

3. (Double coset counting formula). Use (b) to show that

   \[|HaK|=\frac{|H||K|}{|H\cap aKa^{-1}|}. \nonumber \]

[x33c]The normal subgroups \(N\) of a group \(G\) are those with the following property: for every set \(X\) on which \(G\) acts transitively, \(N\) fixes one \(x\) in \(X\) if and only if \(N\) fixes every \(x\) in \(X\).

[x33d](This exercise assumes a knowledge of categories.) Let \(G\) be a group, and let \(F\) be the functor sending a \(G\)-set to its underlying set. We can regard \(G\) as a \(G\)-set, and so an automorphism \(a\) of \(F\) defines an automorphism \(a_{G}\) of \(G\) (as a set). Show that the map \(a\mapsto a_{G}(1)\colon\Aut(F)\rightarrow G\) is an isomorphism of groups (cf. sx66588).