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Normal subgroup of a finite group

Source: Schweitzer 2009

November 13, 2009
group theoryabstract algebrainductiongeometrygeometric transformationlinear algebramatrix

Problem Statement

Let G G be a finite non-commutative group of order t \equal{} 2^nm, where n,m n, m are positive and m m is odd. Prove, that if the group contains an element of order 2n 2^n, then (i) G G is not simple; (ii) G G contains a normal subgroup of order m m.