Number Theory: An Introduction to Mathematics

424 X A Character Study

elements (12),(13),(23) of order 2, andC 3 containing the two elements (123),(132) of
order 3. The irreducible characterψ 1 ofA 3 is self-conjugate and yields two irreducible
characters ofS 3 of degree 1, the trivial characterχ 1 and the sign characterχ 2 =χ 1 λ.
The irreducible charactersψ 2 ,ψ 3 ofA 3 are conjugate and yield a single irreducible
characterχ 3 ofS 3 of degree 2. Thus we obtain the character table:

S 3

C 1 C 2 C 3

χ 1 111

χ 2 1 − 11

χ 3 20 − 1

The elements ofA 4 form four conjugacy classes:C 1 containing only the iden-
tity elemente,C 2 containing the three elementst 1 =( 12 )( 34 ),t 2 =( 13 )( 24 ),t 3 =
( 14 )( 23 )of order 2,C 3 containing four elements of order 3, namelyc,ct 1 ,ct 2 ,ct 3 ,
wherec=( 123 ),andC 4 containing the remaining four elements of order 3, namely
c^2 ,c^2 t 1 ,c^2 t 2 ,c^2 t 3. MoreoverN =C 1 ∪C 2 is a normal subgroup of order 4,H =
{e,c,c^2 }is a cyclic subgroup of order 3, and

A 4 =HN, H∩N={e}.

Ifχis a character of degree 1 ofH, then a characterψof degree 1 ofA 4 is defined by

ψ(hn)=χ(h) for allh∈H,n∈N.

SinceHis isomorphic toA 3 , we obtain in this way three charactersψ 1 ,ψ 2 ,ψ 3 ofA 4
of degree 1. SinceA 4 has order 12, and 12 = 1 + 1 + 1 +9, the remaining
irreducible characterψ 4 ofA 4 has degree 3. The character table ofA 4 can be
completed by means of the orthogonality relations (11) and has the following form,
where againω=(− 1 +i

√

3 )/2.

A 4

|C| 1344

CC 1 C 2 C 3 C 4

ψ 1 1111

ψ 2 11 ωω^2

ψ 3 11 ω^2 ω

ψ 4 3 −10 0

The groupS 4 containsA 4 as a subgroup of index 2 andv=( 12 )∈S 4 \A 4 .The
elements ofS 4 form five conjugacy classes:C 1 containing only the identity element
e,C 2 containing six transpositions (jk)( 1 ≤j<k≤ 4 ),C 3 containing the three
elements of order 2 inA 4 ,C 4 containing eight elements of order 3, andC 5 containing
six elements of order 4.
The self-conjugate characterψ 1 ofA 4 yields two characters ofS 4 of degree 1,
the trivial characterχ 1 and the sign characterχ 2 =χ 1 λ; the pair of conjugate char-
actersψ 2 ,ψ 3 ofA 4 yields an irreducible characterχ 3 ofS 4 of degree 2; and the