Proper and normal in quasisimple implies central
This page describes additional conditions under which a subgroup property implication can be reversed, viz a weaker subgroup property, namely Normal subgroup (?), can be made to imply a stronger subgroup property, namely central subgroup
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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., quasisimple group) must also satisfy the second group property (i.e., group in which every proper normal subgroup is central)
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In other words, a quasisimple group is a group in which every proper normal subgroup is central. This is equivalent to saying that every proper subnormal subgroup is central.
Given: A group with and simple. A normal subgroup of . is the center of .
To prove: or .
- is a normal subgroup of : Since is normal in , fact (2) tells us that the image of under the quotient map is a normal subgroup of . This image is precisely .
- Either or : Since is simple, is either trivial or , so (forcing ) or .
- If , then is abelian: By fact (1), , which is a quotient of abelian groups, hence abelian.
- If , then : Since is abelian, , so , since by assumption.
Steps (2) and (4) complete the proof.