91Ó°ÊÓ

In our proof of Euler's criterion (Theorem 2.21 ), we really only used the fact that \(\mathbb{Z}_{p}^{*}\) has a unique element of multiplicative order \(2 .\) This exercise develops a proof of a generalization of Euler's criterion, based on the fundamental theorem of finite abelian groups. Suppose \(G\) is an abelian group of even order \(n\) that contains a unique element of order 2 . (a) Show that \(G \cong \mathbb{Z}_{2} \times \mathbb{Z}_{m_{1}} \times \cdots \times \mathbb{Z}_{m_{k}},\) where \(e>0\) and the \(m_{i}\) 's are odd integers. (b) Using part (a), show that \(2 G=G\\{n / 2\\}\).

Let \(G\) be a non-trivial, finite abelian group. Let \(s\) be the smallest positive integer such that \(G=\left\langle a_{1}, \ldots, a_{s}\right\rangle\) for some \(a_{1}, \ldots, a_{s} \in G\). Show that \(s\) is equal to the value of \(t\) in Theorem \(6.45 .\) In particular, \(G\) is cyclic if and only if \(t=1\)

Suppose \(G \cong \mathbb{Z}_{m_{1}} \times \cdots \times \mathbb{Z}_{m_{t}} .\) Let \(p\) be a prime, and let \(s\) be the number of \(m_{i}\) 's divisible by \(p\). Show that \(G\\{p\\} \cong \mathbb{Z}_{p}^{\times s}\).

Suppose \(G \cong \mathbb{Z}_{m_{1}} \times \cdots \times \mathbb{Z}_{m_{t}}\) with \(m_{i} \mid m_{i+1}\) for \(i=1, \ldots, t-1\), and that \(H\) is a subgroup of \(G .\) Show that \(H \cong \mathbb{Z}_{n_{1}} \times \cdots \times \mathbb{Z}_{n_{t}},\) where \(n_{i} \mid n_{i+1}\) for \(i=1, \ldots, t-1\) and \(n_{i} \mid m_{i}\) for \(i=1, \ldots, t\)

Suppose that \(G\) is an abelian group such that for all \(m>0\), we have \(m G=G\) and \(|G\\{m\\}|=m^{2}\) (note that \(G\) is not finite). Show that \(G\\{m\\} \cong \mathbb{Z}_{m} \times \mathbb{Z}_{m}\) for all \(m>0 .\)