Chapter 4

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Replace each of the hydrogen atom in a molecule of benzene with a fluorine, chlorine, bromine, or iodine atom.

In the dihedral group \(D_{4}\), find the normalizer of \(P=\\{e, \tau\\}\), where \(\tau\) is a flip. Is \(P\) a normal subgroup of \(D_{4}\) ?

Let \(G\) be any group and \(X\) the set of all subgroups of \(G\). Show that \(X\) is a \(G\) -set under conjugation: \((g, H) \rightarrow g H g^{-1}\).

In Exercises 10 through 12 let \(G\) be a group let \(P(G)\) be the set of all subsets of \(G\), and consider the map \(G \times P(G) \rightarrow P(G)\) sending \((g, A)\) to \(g A g^{-1}=\left\\{g a g^{-1} \mid a \in A\right\\}\). Show that this map is a group action

In Exercises 1 through 11 find the number of essentially different ways in which we can do what is described. Distribute nine balls equally among three children, if there are two white balls, three red balls, and four black balls.

In Exercises 10 through 12 let \(G\) be a group let \(P(G)\) be the set of all subsets of \(G\), and consider the map \(G \times P(G) \rightarrow P(G)\) sending \((g, A)\) to \(g A g^{-1}=\left\\{g a g^{-1} \mid a \in A\right\\}\). Show that the centralizer \(C(A)=\left\\{g \in G \mid g a g^{-1}=a\right.\) for all \(\left.a \in A\right\\}\) is a subgroup of \(G\)

For \(\tau=(12)(34),\) determine all the elements of the centralizer \(C(\tau)\) of \(\tau\) in \(S_{n}\) for \(n \geq 4\)

Let \(G\) be a finite group with \(|G|=n,\) and let \(p\) be the smallest prime dividing \(n\). Show that if \(H\) is a subgroup of \(G\) with \([G: H]=p,\) then \(H \triangleleft G\).

Let \(X\) be a set with four elements. Find the number of equivalence relations on \(X\) that are not equivalent under any permutation in \(S_{4}\).

Let \(H\) be a normal subgroup of \(G\) and \(K\) a normal subgroup of \(H\). Then \(K\) is a subgroup of \(G\). Must \(K\) be a normal subgroup of \(G\) ? If so, give a proof; if not, give a counterexample.

In Exercises 10 through 12 let \(G\) be a group let \(P(G)\) be the set of all subsets of \(G\), and consider the map \(G \times P(G) \rightarrow P(G)\) sending \((g, A)\) to \(g A g^{-1}=\left\\{g a g^{-1} \mid a \in A\right\\}\). Show that the normalizer \(N(A)=\left\\{g \in G \mid g A g^{-1}=A\right\\}\) is a subgroup of \(G\).

In Exercises 12 through 14 for the indicated permutation \(\sigma \in S_{5},\) find \((\) a) the number of conjugates of a in \(S_{5},\) (b) the centralizer of o in \(S_{5},\) (c) the centralizer of o in \(A_{5}\), (d) the number of conjugates of \(\mathrm{\sigma}\) in \(\mathrm{A}_{5}\). $$ \sigma=(524) $$

Let \(X_{1}\) and \(X_{2}\) be \(G\) -sets for the same group \(G,\) and assume \(X_{1} \cap \mathrm{X}_{2}=\varnothing\). Show how \(X_{1} \cup X_{2}\) can be made into a \(G\) -set in a natural way.

Show that no group of order \(p q\) where \(p\) and \(q\) are distinct primes is simple.

In Exercises 15 through 18 let \(H\) be a subgroup of a group \(G\) and let \(X\) be the set \(\\{x H \mid x \in G\\}\) of all left cosets of \(H\) in \(G\). Let \(G\) act on \(H\) by left multiplication \((g, x H) \rightarrow g x H \in X\). Show that this is indeed a group action.

Let \(G\) be a group acting on itself by conjugation. Show that if \(a\) and \(b\) are conjugates in \(G\), then \(|C(a)|=|C(b)|\).

Find two permutations \(\mathrm{O}\) and \(\rho\) in \(S_{5}\) that are conjugates in \(S_{5}\) but not in \(A_{5}\).

Let \(H\) be a normal \(p\) -subgroup of a finite group \(G\). Show that \(H\) is contained in every Sylow \(p\) -subgroup of \(G\).

Let \(H\) be a subgroup of a group \(G\) and let \(X\) be the set \(\\{x H \mid x \in G\\}\) of all left cosets of \(H\) in \(G\). Let \(G\) act on \(H\) by left multiplication \((g, x H) \rightarrow g x H \in X\). Let \(\chi: G \rightarrow S_{X}\) be the permutation representation of the action. Then (a) Determine the kernel \(K\) of \(\chi\). (b) Show that \(K \subset H\) (c) Show that if \(N\) is a normal subgroup of \(G\) and \(N \subset H\), then \(N \subset K\). In other words, show that \(K\) is the largest normal subgroup of \(G\) contained in \(H\).

Let \(G\) be a \(p\) -group and \(H\) a proper subgroup of \(G\). Show that there exists a subgroup \(K \leq G\) such that (a) \(H \leq K\) (b) \(K \triangleleft G\) (c) \([G: K]=p\)

Pertain to Example \(4.7 .16,\) were we have a group \(G\) of order 30, an element \(a\) of order \(2,\) an element \(b\) of order \(15,\) and a relation of form \(b a=a b^{i}\) that holds for some \(i\). Show that if \(b a=a b^{-1}\), then \(G\) is isomorphic to \(D_{15}\)

Show that no group \(G\) of order \(|G|=n\) is simple where (a) \(n=45\) (b) \(n=16\) (c) \(n=p^{r}, p\) prime, \(r>1\) (d) \(n=p^{r} m, p\) prime, \(r \geq 1, p>m\)

Let \(H\) be a subgroup of a group \(G\) and let \(X\) be the set \(\\{x H \mid x \in G\\}\) of all left cosets of \(H\) in \(G\). Let \(G\) act on \(H\) by left multiplication \((g, x H) \rightarrow g x H \in X\). Let \(i=[G: H]\) be the index of \(H\) in \(G\). Then (a) Show that if \(\chi\) is one to one, then \(|G|\) divides \(i !\) (b) Show that if \(|G|\) does not divide \(i !\), then the kernel \(K\) is nontrivial. (c) Show that if \(|G|\) does not divide \(i !,\) then \(G\) has a nontrivial proper normal subgroup.

Let \(r\) be the index \([G: Z(G)]\) of the center \(Z(G)\). Show that for any \(g \in G\), the number of elements \(|K(g)|\) of the conjugacy class \(K(g)\) of the element \(g\) is less than or equal to \(r\).

Explain why for any finite group \(G\) and any \(g \in G,\) the number of elements \(|K(g)|\) of the conjugacy class \(K(g)\) of the element \(g\) divides the order \(1 G\) of the group \(G\).

Find four nonisomorphic groups of order 66 .

Explain why for any \(g \in G,\) the center \(Z(G)\) of the group \(G\) is contained in the centralizer \(C(g)\) of the element \(g\).

Explain why for any \(g \in G,\) the center \(Z(G)\) and the centralizer \(C(g)\) are equal if and only if \(g \in Z(G)\).

Show that for any non-Abelian group \(G,\) the index of the center \([G: Z(G)]\) cannot be a prime \(p\).

Show that for any group \(G,\) the index of the center \([G: Z(G)]\) cannot be a prime \(p\).

Let \(G\) be a group of order 60 . Suppose that \(G\) contains a normal subgroup of order 2. Show that \(G\) has normal subgroups of orders \(6,10,\) and 30 .

By definition, \(a, b \in G\) are conjugate if there exists a \(g \in G\) such that \(b=\mathrm{gag}^{-1}\). Give an example to show that this \(g\) need not be unique or, in other words, that there may be another \(h \neq g\) such that \(b=h a h^{-1}\) also.

Let \(G\) be a finite group, \(P\) a Sylow \(p\) -subgroup of \(G,\) and \(H\) a subgroup of \(G\) with \(P \leq H \leq G\). Show that if \(P\) is normal in \(H\) and \(H\) is normal in \(G,\) then \(P\) is normal in \(G\)

Show that for any element \(g \in G\) not equal to the identity e we have \(|C(g)| \geq 2\).

Show that a group of order 8 is isomorphic to one of the following groups: \(\mathbb{Z}_{8}, \mathbb{Z}_{4} \times \mathbb{Z}_{2}, \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}, D_{4},\) or \(Q_{8}\) the quaternion group