A5 is simple: Difference between revisions

Latest revision as of 17:21, 23 October 2023

Statement

The alternating group of degree five (denoted $A_{5}$ ) is a simple group.

Related facts

A stronger statement about $A_{5}$ : A5 is the simple non-Abelian group of smallest order

Proof

Proof using sizes of conjugacy classes

Further information: element structure of alternating group:A5

The conjugacy class sizes are $1,12,12,20,15$ . A normal subgroup must contain the conjugacy class of size $1$ , and one or more other conjugacy classes. Thus, the order of any normal subgroup must be a sum of some of these numbers, including the $1$ . By Lagrange's theorem, the order must also divide the order of the group. But no such sum among these numbers divides $60$ , other than $1$ and $60$ themselves.

Proof by listing subgroups

Further information: subgroup structure of alternating group:A5

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

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 ==Related facts==
-* [[A5 is the simple non-Abelian group of smallest order]]
+* A stronger statement about <math>A_5</math>: [[A5 is the simple non-Abelian group of smallest order]]
 ==Proof==
 ===Proof using sizes of conjugacy classes===
+{{further|[[element structure of alternating group:A5]]}}
 The conjugacy class sizes are <math>1, 12, 12, 20, 15</math>. A normal subgroup must contain the conjugacy class of size <math>1</math>, and one or more other conjugacy classes. Thus, the order of any normal subgroup must be a sum of some of these numbers, including the <math>1</math>. By [[Lagrange's theorem]], the order must also divide the order of the group. But no such sum among these numbers divides <math>60</math>, other than <math>1</math> and <math>60</math> themselves.
 ===Proof by listing subgroups===
+{{further|[[subgroup structure of alternating group:A5]]}}
 {{fillin}}