The Lattice Structure of the Subgroups of Order 42 and 48 in the Subgroup Lattices of $2 \times 2$ Matrices Over $Z_{7}$

Abstract

Let $\mathcal{G}= \left\{\left( \begin{array}{cc}
a & b \\
c & d \end{array}
\right): a,b,c,d \in Z_{p}, ad-bc \neq 0\right\}$. Then $\mathcal{G}$ is a group under matrix multiplication modulo p. Let $G = \left\{\left( \begin{array}{cc}
a & b \\
c & d \end{array}
\right)\in \mathcal{G} : ad-bc = 1\right\}$. Then G is a subgroup of $\mathcal{G}$. We have, $o(\mathcal{G}) = p(p^{2}-1)(p-1)$ and $o(G) = p(p^{2}-1)$. Let $L(G)$ denotes the lattice of subgroups of G, where G is the group of $2\times 2$ matrices over $Z_p$ having determinant value 1 under matrix multiplication modulo p, where p is a prime number. In this paper, we give the structure of the subgroups of order 42 and 48 of $L (G)$ in the case when $p=7$.