Cocentrality is transitive

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of a subgroup property (i.e., cocentral subgroup) satisfying a subgroup metaproperty (i.e., 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 cocentral subgroup |Get facts that use property satisfaction of cocentral subgroup | Get facts that use property satisfaction of cocentral subgroup|Get more facts about transitive subgroup property


Statement

Statement with symbols

If H is a cocentral subgroup of K, and K is a cocentral subgroup of G, then H is a cocentral subgroup of G.

Definitions used

Cocentral subgroup

Further information: Cocentral subgroup

A subgroup H of a group G is termed cocentral in G if HZ(G) = G, where Z(G) is the center of G.

Related facts

Proof

Given: A group G, a cocentral subgroup K of G, a cocentral subgroup H of K. In other words, HZ(K) = K, and KZ(G) = G.

To prove: H is cocentral in G, i.e., HZ(G) = G.

Proof: Consider C_G(Z(K)). Clearly, Z(G) \le C_G(Z(K)), and K \le C_G(Z(K)), because Z(K) centralizes K. Thus, KZ(G) \le C_G(Z(K)), so C_G(Z(K)) = G. Thus, Z(K) \le Z(G). Thus, HZ(K) \le HZ(G), so K \le HZ(G), and thus G = KZ(G) \le HZ(G). Thus, <mathG = HZ(G)</math>.