Self-centralizing and minimal normal implies fully invariant in co-Hopfian group

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Statement

Suppose G is a Co-Hopfian group (?) and H is a Minimal normal subgroup (?) of G that is also a Self-centralizing subgroup (?), i.e., C_G(H) \le H. Then, H is a Fully invariant subgroup (?) of G.

Facts used

  1. Self-centralizing and minimal normal implies monolith
  2. Monolith is fully invariant in co-Hopfian group

Proof

The proof follows directly from facts (1) and (2).