Math 541
Solution to HW #9
Assignment: Prove that if
G
is a simple group of size less than 60, then
G
is cyclic of prime size.
Recall from class the following lemmas:
•
Lemma P: If

G

=
p
with
p
a prime, then
G
is a simple group.
•
Lemma Z: If

G

=
p
a
with
p
a prime and
a
an integer greater than 1, then
G
is not a simple group.
•
Lemma S: If

G

is not a prime, and
G
has a singleton conjugacy class not equal to
{
e
}
, then
G
is
not simple.
•
Lemma r!: If
G
has a conjugacy class of size
r >
1 and

G

> r
!, then
G
is not simple.
•
Super Lemma r!: If
G
has a conjugacy class of size
r >
1 with

G

< r
!, and

G

does not divide
r
!,
then
G
is not simple.
As a consequence of the above lemmas, we can add the following:
•
Lemma PQ: If

G

=
pq
with
p
and
q
both primes, then
G
is not a simple group.
–
Proof
: Seeking a contradiction, let
G
be a group of order
pq
that is simple. By Lemma S,
G
has no singleton conjugacy class besides the identity. Since the size of any conjugacy class of
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 Fall '09
 Pollack
 Math, Algebra, Prime number, US standard clothing size, Lemma, singleton conjugacy

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