Characteristicity is not finite-relative-intersection-closed

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) not satisfying a subgroup metaproperty (i.e., finite-relative-intersection-closed subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about characteristic subgroup|Get more facts about finite-relative-intersection-closed subgroup property|

Statement

Verbal statement

It is possible to have a group G with subgroups H,K,L, such that H \le L, K \le L, H characteristic in G and K characteristic in L, but H \cap K not characteristic in G.

Related facts

Other facts about finite-relative-intersection-closed

Related metaproperty satisfactions and dissatisfactions for characteristicity

Example

Example where L = HK

Further information: symmetric group:S4, dihedral group:D8, subgroup structure of symmetric group:S4

Suppose G is the symmetric group on the set \{ 1,2,3,4 \}. Define:

H := \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}, K := \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2) \}

and define L = HK, explicitly, L is a 2-Sylow subgroup of G isomorphic to the dihedral group:

L := \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2), (1,2)(3,4), (1,4)(2,3), (1,3), (2,4) \}.

Then:

  • H is characteristic in G: In fact, it is the only normal subgroup of order 4 in G, and also equals the second commutator subgroup of G.
  • K is characteristic in L: In fact, K is an isomorph-free subgroup of L.
  • H \cap K is not characteristic in G: The intersection H \cap K is the two-element subgroup \{ (), (1,3)(2,4)\}, which is not characteristic -- in fact, it is not even normal in G, as it is not invariant under conjugation by (2,3).

Example where K \le H

Further information: symmetric group:S4, dihedral group:D8, subgroup structure of symmetric group:S4

Suppose G is the symmetric group on the set \{ 1,2,3,4 \}. Define:

H := \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}, K := \{ (), (1,3)(2,4) \}

and L is a 2-Sylow subgroup of G isomorphic to the dihedral group, containing H:

L := \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2), (1,2)(3,4), (1,4)(2,3), (1,3), (2,4) \}.

Then:

  • H is characteristic in G: In fact, it is the only normal subgroup of order 4 in G, and also equals the second commutator subgroup of G.
  • K is characteristic in L: In fact, K is the center of L.
  • H \cap K is not characteristic in G: The intersection H \cap K is the two-element subgroup K = \{ (), (1,3)(2,4)\}, which is not characteristic -- in fact, it is not even normal in G, as it is not invariant under conjugation by (2,3).