Difference between revisions of "Hall not implies order-isomorphic"

From Groupprops
Jump to: navigation, search
Line 6: Line 6:
 
==Statement==
 
==Statement==
  
Two [[Hall subgroup]]s of the same order in a group, need not be isomorphic.
+
Two [[Hall subgroup]]s of the same order in a [[finite group]], need not be isomorphic.
  
 
==Proof==
 
==Proof==

Revision as of 10:58, 8 August 2008

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., Hall subgroup) need not satisfy the second subgroup property (i.e., order-isomorphic subgroup)
View all subgroup property non-implications | View all subgroup property implications

Statement

Two Hall subgroups of the same order in a finite group, need not be isomorphic.

Proof

An example is the group SL(2,11). This group has order 2^2.3.11, and the 11'-Hall subgroups have order 12. It turns out that there are two possible isomorphism classes of such subgroups: the dihedral group on twelve elements, and the alternating group on four elements.

Here are explicit embeddings of these subgroups:

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]