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Properties of a class of groups on \(\mathbb{Z}_m\). (English) Zbl 1389.20010

Summary: In this paper, we obtain the factorization of direct production and order of group \({\text{GL}}\left(n, \mathbb Z_m\right)\) in a simple method. Then we generalize some properties of \({\text{GL}}\left(2, \mathbb Z_p \right)\) proposed in a literature, and prove that the group \({\text{GL}}\left(2, \mathbb Z_{z^\lambda} \right)\) is solvable. We also prove that group \({\text{GL}}\left(n, \mathbb Z_p \right)\) is solvable if and only if \({\text{GL}}\left(n, \mathbb Z_p \right)\) is solvable, and list the generators of groups \({\text{GL}}\left(n, \mathbb Z_p \right)\) and \({\text{SL}}\left(n, \mathbb Z_p \right)\). At last, we prove that \({\text{PSL}}\left( 2,\mathbb Z_p \right)\left( {p > 3} \right)\) and \({\text{PSL}}\left(n, \mathbb Z_p \right)\left(n > 3 \right)\) are simple.

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D40 Products of subgroups of abstract finite groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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References:

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