Vipul: New page: {{subgroup property implication| stronger = cocentral subgroup| weaker = centralizer-dense subgroup}} ==Definitions used== ===Cocentral subgroup=== {{further|cocentral subgroup}} A...

2008-09-21T20:41:34Z

New page: {{subgroup property implication| stronger = cocentral subgroup| weaker = centralizer-dense subgroup}} ==Definitions used== ===Cocentral subgroup=== {{further|cocentral subgroup}} A...

New page

{{subgroup property implication|
stronger = cocentral subgroup|
weaker = centralizer-dense subgroup}}

==Definitions used==

===Cocentral subgroup===

{{further|[[cocentral subgroup]]}}

A subgroup <math>H</math> of a group <math>G</math> is termed '''cocentral''' if <math>HZ(G) = G</math> where <math>Z(G)</math> denotes the [[center]] of <math>G</math>.

==Proof==

'''Given''':A [[group]] <math>G</math> with center <math>Z(G)</math>, a cocentral subgroup <math>H</math>. In other words, <math>HZ(G) = G</math>.

'''To prove''': <math>C_G(H) = Z(G)</math>, where <math>C_G(H)</math> denotes the [[centralizer]] of <math>H</math> in <math>G</math>.

'''Proof''': Suppose <math>K = C_G(H)</math>. Then <math>H \le C_G(K)</math>. Also, <math>Z(G) \le C_G(K)</math>. Thus, <math>HZ(G) \le C_G(K)</math>, so <math>G \le C_G(K)</math>, forcing <math>C_G(K) = G</math>, forcing <math>K \le Z(G)</math>. On the other hand, <math>Z(G) \le K</math> since anything in the center of <math>G</math> centralizes <math>H</math>. Thus, <math>K = Z(G)</math>, completing the proof.

Cocentral implies centralizer-dense - Revision history

Vipul: New page: {{subgroup property implication| stronger = cocentral subgroup| weaker = centralizer-dense subgroup}} ==Definitions used== ===Cocentral subgroup=== {{further|cocentral subgroup}} A...