Every subgroup is contracharacteristic in its normal closure

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This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Contracharacteristic subgroup (?) and Normal subgroup (?)), to another known subgroup property (i.e., Subgroup (?))
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This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
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Statement

Any subgroup of a group is a contracharacteristic subgroup of its normal closure. In particular, it occurs as a contracharacteristic subgroup of a normal subgroup.

Definitions used

Contracharacteristic subgroup

Further information: Contracharacteristic subgroup

A subgroup of a group is termed contracharacteristic if it is not contained in any proper characteristic subgroup.

Related facts

Facts used

  1. Characteristic of normal implies normal

Proof

Hands-on proof

Given: A subgroup H \le G, H^G is the normal closure of H in G.

To prove: If K \le H^G is a characteristic subgroup of H^G containing H, then K = H^G.

Proof:

  1. By fact (1), we see that since K is characteristic in H^G and H^G is normal in G, we obtain that K is normal in G.
  2. Thus, K is a normal subgroup of G containing H. By definition of normal closure, we get that H^G \le K. Since K \le H^G, we get K = H^G.