Show that \(a=1891\) is relatively prime to \(n=3797\) and determine the multiplicative inverse of \(\bar{a}\) in \(\mathbb{Z}/n\mathbb{Z}\).

Prove that if \(x^2=1\) for all \(x \in G\), then \(G\) is abelian.

If \(n=2k\) is even and \(n \ge 4\), show that \(z=r^k\) is an element of order \(2\) which commutes with all elements of \(D_{2n}\). Show also that \(z\) is the only nonidentity element of \(D_{2n}\) that commutes with all elements of \(D_{2n}\).

Prove that the order of an element in \(S_n\) equals the least common multiple of the lengths of the cycles in its cycle decomposition.

If \(F\) is a finite field of order \(q\), prove that \[|\operatorname{GL}_n(F)|=\prod_{i=0}^{n-1} \left(q^n-q^i\right).\] Hint: count the number of possibities for the first column, then the second, … What conditions are needed at each step for the final matrix to be invertible?

Let \(G\) be any group. Prove that the map from \(G\) to itself defined by \(g \mapsto g^{-1}\) is a homomorphism if and only if \(G\) is abelian.

Prove that \(D_8\) and \(Q_8\) are not isomorphic.

Let \(A\) be a nonempty set and let \(k\) be a positive integer with \(k \le |A|\). The symmetric group \(S_A\) acts on the set \(B\) consisting of all subsets of \(A\) of cardinality \(k\) by \(\sigma \cdot \{a_1,\ldots, a_k\} = \{ \sigma(a_1), \ldots, \sigma(a_k)\}\).

1. Prove that this is a group action.
2. Describe explicity how the elements \((1\ 2)\) and \((1\ 2\ 3)\) act on the six \(2\)-element subsets of \(\{1,2,3,4\}\).