COUNTING THE NUMBER OF SUBGROUPS AND NORMAL SUBGROUPS OF THE GROUP U2np, p IS AN ODD PRIME

Abstract

The aim of  this paper is to compute the number of subgroups and  normal subgroups of the group $U_{2np}=\langle a,b\mid a^{2n}=b^{p}=e, aba^{-1}=b^{-1} \rangle$, where $p$ is an odd prime. Suppose $n=2^r\prod_{1 \leq i \leq s}p_{i}^{\alpha_{i}}$ in which $p_i$'s are distinct odd primes, $\alpha_i$'s are positive integers and $t=\prod_{1 \leq i \leq s}p_{i}^{\alpha_{i}}$. It is proved that the number of subgroups is $2\tau(2n)+(p-1)\left(\tau(\frac{n}{p})+\tau(\frac{n}{2^r}) \right)$, when $p \mid n$ and $2\tau(2n)+(p-1) \left[ \tau(t)\right]$, otherwise. It will be also proved that this group has $\tau(2n)+\tau(n)$ normal subgroups.