# The alternating group on five elements is simple

The al­ter­nat­ing group $$A_5$$ on five el­e­ments is sim­ple.

# Proof

Re­call that $$A_5$$ has or­der $$60$$, so La­grange’s the­o­rem states that any sub­group of $$A_5$$ has or­der di­vid­ing $$60$$.

Sup­pose $$H$$ is a nor­mal sub­group of $$A_5$$, which is not the triv­ial sub­group $$\{ e \}$$. If $$H$$ has or­der di­visi­ble by $$3$$, then by Cauchy’s the­o­rem there is a $$3$$-cy­cle in $$H$$ (be­cause the $$3$$-cy­cles are the only el­e­ments with or­der $$3$$ in $$A_5$$). Be­cause $$H$$ is a union of con­ju­gacy classes, and be­cause the $$3$$-cy­cles form a con­ju­gacy class in $$A_n$$ for $$n > 4$$, $$H$$ would there­fore con­tain ev­ery $$3$$-cy­cle; but then it would be the en­tire al­ter­nat­ing group.

If in­stead $$H$$ has or­der di­visi­ble by $$2$$, then there is a dou­ble trans­po­si­tion such as $$(12)(34)$$ in $$H$$, since these are the only el­e­ments of or­der $$2$$ in $$A_5$$. But then $$H$$ con­tains the en­tire con­ju­gacy class so it con­tains ev­ery dou­ble trans­po­si­tion; in par­tic­u­lar, it con­tains $$(12)(34)$$ and $$(15)(34)$$, so it con­tains $$(15)(34)(12)(34) = (125)$$. Hence as be­fore $$H$$ con­tains ev­ery $$3$$-cy­cle so is the en­tire al­ter­nat­ing group.

So $$H$$ must have or­der ex­actly $$5$$, by La­grange’s the­o­rem; so it con­tains an el­e­ment of or­der $$5$$ since prime or­der groups are cyclic.

The only such el­e­ments of $$A_n$$ are $$5$$-cy­cles; but the con­ju­gacy class of a $$5$$-cy­cle is of size $$12$$, which is too big to fit in $$H$$ which has size $$5$$.

Children:

Parents:

• Alternating group

The al­ter­nat­ing group is the only nor­mal sub­group of the sym­met­ric group (on five or more gen­er­a­tors).

• This is a test comment

• This is definitely a page which ad­mits two lenses: the “easy” proof and the “the­ory-heavy” proof. What kind of lens de­sign might peo­ple use?

• Most tech­ni­cal ver­sion goes onto the pri­mary page (this one). Easier ver­sions get their own lenses. You could ti­tle the lens “Easy proof” or “Sim­ple proof’.