Suppose that \(\star \) is an associative binary operation on a set \(S\). Let

Show that \(H\) is closed under \(\star \). (We think of \(H\) as consisting of all elements of \(S\) that commute with every element in \(S\).)

Are all groups of 4 elements commutative?

Show that if \(G\) is a finite group with identity \(e\) and with an even number of elements, then there is \(a \neq e\) in \(G\) such that \(a \star a = e\).

Show that if \(H\) and \(K\) are subgroups of an abelian group \(G\), then

is a subgroup of \(G\).

Show that a group with no proper nontrivial subgroups is cyclic.

Let \(p\) be a prime number. Find the number of generators of the cyclic group \(\mathbb {Z}_{p^r}\), where \(r\) is an integer \(\geq 1\).

Let \(G\) be an abelian group and let \(H\) and \(K\) be finite cyclic subgroups with \(|H| = r\) and \(|K| = s\).

1. Show that if \(r\) and \(s\) are relatively prime, then \(G\) contains a cyclic subgroup of order \(rs\).

2. Generalizing part (a), show that \(G\) contains a cyclic subgroup of order the least common multiple of \(r\) and \(s\).

Show that for \(n \geq 3\), there exists a nonabelian group with \(2n\) elements that is generated by two elements of order 2.

Find the number of elements in the set \(\{ \sigma \in S_4 \mid \sigma (3) = 3\} \).

Show that \(S_n\) is a nonabelian group for \(n \geq 3\).

Let \(G\) be a group. Prove that the permutations \(\rho _a : G \to G\), where \(\rho _a(x) = xa\) for \(a \in G\) and \(x \in G\), do form a group isomorphic to \(G\).

Let \(G\) be a group and let \(a\) be a fixed element of \(G\). Show that the map \(\lambda _a : G \to G\), given by \(\lambda _a(g) = ag\) for \(g \in G\), is a permutation of the set \(G\).

Show that \(S_n\) is generated by \(\{ (1, 2), (1, 2, 3, \ldots , n)\} \). [Hint: Show that as \(r\) varies, \((1, 2, 3, \ldots , n)^r(1, 2)(1, 2, 3, \ldots , n)^{n-r}\) gives all the transpositions \((1, 2), (2, 3), (3, 4), \ldots , (n-1, n), (n, 1)\). Then show that any transposition is a product of some of these transpositions and use Corollary 9.12.]

Let \(G\) be a group of order \(pq\), where \(p\) and \(q\) are prime numbers. Show that every proper subgroup of \(G\) is cyclic.

Show that a finite cyclic group of order \(n\) has exactly one subgroup of each order \(d\) dividing \(n\), and that these are all the subgroups it has.

Let \(H\) and \(K\) be groups and let \(G = H \times K\). Recall that both \(H\) and \(K\) appear as subgroups of \(G\) in a natural way. Show that these subgroups \(H\) (actually \(H \times \{ e\} \)) and \(K\) (actually \(\{ e\} \times K\)) have the following properties:

1. Every element of \(G\) is of the form \(hk\) for some \(h \in H\) and \(k \in K\).

2. \(hk = kh\) for all \(h \in H\) and \(k \in K\).

3. \(H \cap K = \{ e\} \).

Let \(H\) and \(K\) be subgroups of a group \(G\) satisfying the three properties listed in the preceding exercise. Show that for each \(g \in G\), the expression \(g = hk\) for \(h \in H\) and \(k \in K\) is unique. Then let each \(g\) be renamed \((h, k)\). Show that, under this renaming, \(G\) becomes structurally identical (isomorphic) to \(H \times K\).

Show that a finite abelian group is not cyclic if and only if it contains a subgroup isomorphic to \(\mathbb {Z}_p \times \mathbb {Z}_p\) for some prime \(p\).

Example: Let \(S_n\) be the symmetric group on \(n\) letters, and let \(\Phi : S_n \to \mathbb {Z}_2\) be defined by \(\Phi (\sigma ) = 0\) if \(\sigma \) is an even permutation, 1 if \(\sigma \) is an odd permutation. Show that \(\Phi \) is a homomorphism.

Solution: We must show that \(\Phi (\sigma \mu ) = \Phi (\sigma ) + \Phi (\mu )\) for all choices of \(\sigma , \mu \in S_n\). Note that the operation on the right-hand side of this equation is written additively since it takes place in the group \(\mathbb {Z}_2\). Verifying this equation amounts to checking just four cases:

1. \(\sigma \) odd and \(\mu \) odd,

2. \(\sigma \) odd and \(\mu \) even,

3. \(\sigma \) even and \(\mu \) odd,

4. \(\sigma \) even and \(\mu \) even.