Join of characteristic and characteristic-potentially characteristic implies characteristic-potentially characteristic

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This article describes a computation relating the result of the Join operator (?) on two known subgroup properties (i.e., Characteristic subgroup (?) and Characteristic-potentially characteristic subgroup (?)), to another known subgroup property (i.e., Characteristic-potentially characteristic subgroup (?))
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Statement

Suppose H,K are subgroups of a group G such that H is a characteristic subgroup of G and K is a characteristic-potentially characteristic subgroup of G. Then, the join of subgroups \langle H, K \rangle (which in this case is equal to the product of subgroups HK) is also a characteristic-potentially characteristic subgroup.

Definitions used

Characteristic-potentially characteristic subgroup

Further information: Characteristic-potentially characteristic subgroup

Related facts

Similar facts

About joins:

About intersections:

About composition (subgroups of subgroups):

Facts used

  1. Characteristicity is transitive
  2. Characteristicity is strongly join-closed

Proof

Given: A group G, subgroups H,K of G such that H is characteristic in G and K is characteristic-potentially characteristic in G.

To prove: The join HK is also characteristic-potentially characteristic in G.

Proof: By the definition of characteristic-potentially characteristic, there is a group L containing G such that both K and G are characteristic in L.

  1. H is characteristic in L: H is characteristic in G and G is characteristic in L, so fact (1) yields that H is characteristic in L.
  2. HK is characteristic in L: H and K are both characteristic in L, so by fact (2), so is HK.

Thus, L is a group containing G such that both HK and G are characteristic in L. Thus, HK is characteristic-potentially characteristic in G.