Abstract:
In this paper there is given a formula for the number of solutions of the equation $x^{p^n}=1$ in an arbitrary finite $p$-group $G$ (of exponent $p^l$, $1\le n\le l$) and a formula for the number of cyclic subgroups of $G$ of any order. A connection is established among $|G|$, $p^l$, and the ranks of those subgroups of $G$ of order greater than $p^l$; if $G$ is regular, there are analogous relations among the orders of the characteristic subgroups $\Omega_n=\langle x\mid x\in G,x^{p^n}=1\rangle$, $n=1,2,\dots,l$, and the ranks of the subgroups of $G$ of order greater than $p^n$. These results are precise; some of them strengthen the well-known classical theorems of Frobenius and Miller for $p$-groups.
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