Monolith is strictly characteristic

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., monolith) must also satisfy the second subgroup property (i.e., strictly characteristic subgroup)
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Statement

Verbal statement

If a group has a monolith (a Minimal normal subgroup (?) contained in every nontrivial normal subgroup), then that monolith is a strictly characteristic subgroup (it is invariant under any surjective endomorphism of the group).

Related facts

Facts used

Proof

Given: A group G, a minimal normal subgroup N such that N \le M for any nontrivial normal subgroup M. A surjective endomorphism \sigma of G.

To prove: \sigma(N) \le N.

Proof: Consider the subgroup \sigma^{-1}(N). This is normal by fact (1), either \sigma^{-1}(N) is trivial or N \le \sigma^{-1}(N). Since \sigma is surjective and N is nontrivial, \sigma^{-1}(N) cannot be trivial. Thus, N \le \sigma^{-1}(N). This forces that \sigma(N) \le N, as desired.