Finite direct power-closed characteristic is quotient-transitive

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of a subgroup property (i.e., finite direct power-closed characteristic subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about finite direct power-closed characteristic subgroup |Get facts that use property satisfaction of finite direct power-closed characteristic subgroup | Get facts that use property satisfaction of finite direct power-closed characteristic subgroup|Get more facts about quotient-transitive subgroup property


Statement

Statement with symbols

Suppose G is a group and H and K are subgroups with H \le K \le G. Suppose H is a finite direct power-closed characteristic subgroup of G. Since characteristic implies normal, we can talk of the quotient group G/H. Suppose, further, that K/H is a finite direct power-closed characteristic subgroup of G/H. Then, K is also a finite direct power-closed characteristic subgroup of G.

Facts used

  1. Characteristicity is quotient-transitive: If A \le B \le C with A characteristic in C and B/A characteristic in C/A, then B is characteristic in C.
  2. (no link): The fact that (K/H)^n \cong K^n/H^n and (G/H)^n \cong G^n/H^n, and the natural embedding from (K/H)^n to (G/H)^n coincides via these isomorphisms with the natural embedding from K^n/H^n to G^n/H^n. where {}^n stands for the n^{th} direct power.

Proof

Given: A group G, a finite direct power-closed characteristic subgroup H of G. A subgroup K of G containing H such that K/H is a finite direct power-closed characteristic subgroup of G/H. n is a natural number.

To prove: K^n is a characteristic subgroup of G^n under the natural embedding.

Proof:

Step no. Assertion Definitions used Facts used Given data used Previous steps used Explanation
1 (K/H)^n is characteristic in (G/H)^n finite direct power-closed characteristic subgroup -- K/H is finite direct power-closed characteristic in G/H -- Piece together definition and given data.
2 K^n/H^n is characteristic in G^n/H^n -- fact (2) -- step (1) Using the natural identification indicated by fact (2), step (2) becomes a reformulation of step (1).
3 H^n is characteristic in G^n finite direct power-closed characteristic subgroup -- H is finite direct power-closed characteristic in G -- Piece together definition and given data.
4 K^n is characteristic in G^n -- fact (1) -- steps (2) and (3) Piece together. [SHOW MORE]