Self-centralizing and minimal normal implies monolith

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Statement

Any Minimal normal subgroup (?) of a group that is self-centralizing, is a Monolith (?): it is the unique minimal normal subgroup, and is contained in every minimal normal subgroup.

Related facts

Applications

Facts used

  1. Self-centralizing and normal implies normality-large
  2. Normality-large and minimal normal implies monolith

Proof

The proof follows directly by combining facts (1) and (2).