Alternating group:A4

From Groupprops
Revision as of 11:27, 11 May 2007 by Vipul (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]


Permutation definition

The alternating group A_4 is defined as the group of all even permutations on a set of 4 elements.


Group properties


This particular group is solvable

The commutator subgroup of A_4 is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.

Thus, A_4 is solvable of solvable length 2, or in other words, it is a metabelian group.


This particular group is not nilpotent


This particular group is not Abelian


This particular group is not simple

Since A_4 has a proper nontrivial commutator subgroup, it is not simple.