Chapter 2

Let \(G\) be a group of order \(36 .\) If \(G\) has an element \(a \in G\) such that \(a^{12} \neq e\) and \(a^{18} \neq e,\) show that \(G\) is cyclic.

Determine whether the indicated subgroup is normal in the indicated group. Show that if \(H \triangleleft G\) and \(K \triangleleft G\), then \(H K \triangleleft G\).

Find all possible nontrivial homomorphisms between the indicated groups. $$ \phi: \mathbb{Z}_{10} \rightarrow \mathbb{Z}_{5} $$

A subgroup \(H\) of a group \(G\) is called a characteristic subgroup of \(G\) if for all \(\phi \in \operatorname{Aut}(G)\) we have \(\phi(H)=H\). Let \(G\) be a group, \(H\) a characteristic subgroup of \(G,\) and \(K\) a characteristic subgroup of \(H\). Show that \(K\) is a characteristic subgroup of \(G\).

Let \(\phi: G \rightarrow G^{\prime}\) be an onto homomorphism with Kern \(\phi=K,\) and let \(H^{\prime}\) be a subgroup of \(G^{\prime}\). Show that there exists a subgroup \(H\) of \(G\) such that \(K \subseteq H\) and \(H / K \propto H^{\prime}\)

I.et \(H\) and \(K\) be subgroups of a group \(G\) such that \(H

Show that \(\operatorname{Aut}\left(D_{4}\right) \cong D_{4}\)

Let \(G\) be a group with \(|G|=p^{2}\), where \(p\) is prime. Show that every proper subgroup of \(G\) is cyclic.

Consider the dihedral group \(D_{6}=\left\\{\rho^{i} \tau^{j} \mid 0 \leq i<6,0 \leq j<2\right\\},\) where \(\rho^{6}=\tau^{2}=\) identity, and \(\rho \tau=\tau \rho^{-1}\). Show that (a) \(\left\langle\rho^{3}\right\rangle

Show that \(\operatorname{Aut}\left(Q_{8}\right) \cong S_{4}\)

Let \(G\) be a group with \(|G|=p q,\) where \(p\) and \(q\) are primes. Show that every proper subgroup of \(G\) is cyclic.

Find the normalizer of the indicated subgroup in the indicated group. $$ A_{3} \text { in } S_{3} $$

Consider the dihedral group \(D_{n}=\left\\{\rho^{i} \tau^{j} \mid 0 \leq i

Show that \(\operatorname{Aut}\left(S_{3}\right) \cong S_{3}\)

Let \(H\) and \(K\) be subgroups of a group \(G\), with \(|H|=n\) and \(\mid K\\}=m\), where \(\operatorname{gcd}(n, m)=1 .\) Show that \(H \cap K=\\{e\\} .\)

Consider the relation \(R\) on the class of all groups defined by the condition that \(G R G^{\prime}\) if and only if \(G\) and \(G^{\prime}\) are isomorphic. Show that \(R\) has the properties of an equivalence relation (reflexivity, symmetry, transitivity).

Find the normalizer of the indicated subgroup in the indicated group. $$ \left\langle\mu_{1}\right\rangle \text { in } S_{3} $$

Let \(Z(G)\) be the center of a group \(G\). Show that (a) \(Z(G)

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi: S_{3} \rightarrow S_{5} $$

Show that \(n^{19}-n\) is divisible by 21 for any integer \(n\).

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi: \mathbb{Z}_{3} \rightarrow \mathbb{Z}_{5} $$

Find the normalizer of the indicated subgroup in the indicated group. $$ \langle\mathrm{j}\rangle \text { in } Q_{8} $$

Let \(G\) be a group and \(S\) a subset of \(G\). Define \(\langle S\rangle\) to be the smallest subgroup of \(G\) containing \(S\), called the subgroup of \(G\) generated by \(S .\) Show that \(\langle S\rangle\) exists.

Find the remainder of \(9^{1573}\) when divided by 11 .

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi: \mathbb{Z}_{4} \rightarrow \mathbb{Z}_{12} $$

Let \(H\) and \(K\) be subgroups of a group \(G\). Show that \(H K\) is a subgroup of \(G\) if and only if \(H K=K H\).

Show that in \(S_{4}\) the subgroup generated by \\{(12),(1234)\\} (in the sense of the preceding Exercise 25 ) is the whole group: \(\langle(12),(1234)\rangle=S_{4}\).

Compute \(\phi\left(p^{2}\right),\) where \(p\) is prime.

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi: \mathbb{Z}_{5} \rightarrow \mathbb{Z}_{12} $$

Let \(G\) be a group of order \(p q,\) where \(p\) and \(q\) are distinct primes. Suppose \(G\) has a unique subgroup of order \(p\) and a unique subgroup of order \(q\). Show that \(G\) is cyclic,

Let \(G\) be a group and let \(S=\left\\{x y x^{1} y^{-1} \mid x, y \in G\right\\} .\) Let \(N\) be the subgroup \(\langle S\rangle\) generated by \(S\) (in the sense of Exercise 25 ), called the commutator subgroup of \(G\). Show that (a) \(N \triangleleft G\) (b) \(G / N\) is Abelian. (c) If \(H\) is a normal subgroup of \(G\) and \(G / H\) is Abelian, then \(N \subseteq H\). (d) If \(H\) is a subgroup of \(G\) with \(N \subseteq H\), then \(H \triangleleft G\).

Compute \(\phi(p q)\), where \(p\) and \(q\) are distinct primes.

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi: D_{4} \rightarrow S_{5} $$

Let \(G\) be a group with a unique subgroup of order \(n\) and a unique subgroup of order \(m,\) where the positive integers \(n\) and \(m\) are relatively prime. Show that \(G\) has a normal subgroup of order \(n m\).

Find the remainder of \(5^{1258}\) when divided by 12 .

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi: \mathbb{Z} \rightarrow \mathbb{Z}_{7} $$

Let \(K \triangleleft G\) and let \(H\) be a subgroup of \(G\). Show that \(K \cap H \triangleleft H\).

Let \(G\) be a non-Abelian group with \(|G|=2 p\), where \(p\) is prime. Show that there exists a \(g \in G\) such that \(|g|=p\).

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi:{Z}_{10} \rightarrow \mathbb{Z}_{8} $$

In Exercises 23 through 30 construct an example of a nontrivial homomorphism between the two indicated groups, if this is possible, or explain why this is not possible. $$ \phi: S_{5} \rightarrow \mathbb{Z}_{2} $$

Construct Cayley digraphs of the indicated group \(G\) with the indicated generating set \(S\), and specify the defining relations. $$ G=\mathbb{Z}_{6}, \quad S=\\{1\\} $$

Let \(G\) be a group with \(|G|>1\) such that \(G\) has no nontrivial proper subgroups. Show that \(G\) is a finite cyclic group of prime order.

Construct Cayley digraphs of the indicated group \(G\) with the indicated generating set \(S\), and specify the defining relations. $$ G=\mathbb{Z}_{6} \quad S=\\{2,3\\} $$

Let \(G\) be a group of order 15 . Show that \(G\) contains an element of order \(3 .\)

Let \(\phi: Z_{12} \rightarrow \mathbb{Z}_{3}\) be a homomorphism with Kern \(\phi=\\{0,3,6,9\\}\) and \(\phi(4)=2\). Find all the elcments \(x \in \mathbb{Z}_{12}\) such that \(\phi(x)=1,\) and show that they form a coset of Kern \(\phi\) in \(Z_{12}\).

Construct Cayley digraphs of the indicated group \(G\) with the indicated generating set \(S\), and specify the defining relations. $$ G=S_{3} \quad S=\\{(12),(23)\\} $$

Let \(H\) be a subgroup of a finite group \(G\) and \(K\) a subgroup of \(H\). Suppose that the index \([G: H]=n\) and the index \([H: K]=m .\) Show that the index \([G: K]=n m\). (Hint: Let \(x_{i} H\) be the distinct left cosets of \(H\) in \(G\) and \(y_{j} K\) the distinct left cosets of \(K\) in \(H\). Show that \(x_{i} y_{j} K\) are the distinct left cosets of \(K\) in \(G\).)

Construct Cayley digraphs of the indicated group \(G\) with the indicated generating set \(S\), and specify the defining relations. $$ G=D_{4} \quad S=\\{\rho, \tau\\} $$

Let \(H\) and \(K\) be subgroups of a group \(G\) and for all \(a, b \in G\) let \(a \sim b\) if and only if \(a=h b k\) for some \(h \in H\) and \(k \in K\). Show that the relation \(\sim\) so defined is an equivalence relation. Describe the equivalence classes (which are called double cosets).

Let \(\phi: G \rightarrow G^{\prime}\) be a homomorphism, \(K=\) Kern \(\phi\), and \(a \in G\). Show that \(\\{x \in G\) \(\phi(x)=\phi(a)\\}=a K,\) the left coset of \(K\) to which the element \(a\) belongs.