Lemma 7.7.1. If n
3,
then every permutation in An
can be expressed as a product of 3-cycles.
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Lemma 7.7.1.
If n 3,
then every permutation in An
can be expressed as a product of 3-cycles.
Theorem 7.7.2.
The symmetric group Sn is not solvable for
n 5.
Lemma 7.7.3.
If n 4,
then no proper normal subgroup of An contains a 3-cycle.
Theorem 7.7.4.
The alternating group An is simple if
n 5.
Definition 7.7.5.
Let F be a field. The set of all n × n
matrices with entries in F and determinant 1 is called the
special linear group
over F, and is denoted by SLn(F).
The group SLn(F) modulo its center is called the
projective special linear group
and is denoted by PSLn(F).
Proposition 7.7.6.
For any field F, the center of SLn(F)
is the set of nonzero scalar matrices with determinant 1.
Example. 7.7.1.
PSL2(F) 
S3
if |F| = 2.
Example. 7.7.2.
PSL2(F)
A4 if |F| = 3.
Lemma 7.7.7.
Let F be any field. Then SL2(F)
is generated by elements of the form
and
.
Lemma 7.7.8.
Let F be any finite field,
and let N be a normal subgroup of SL2(F).
If N contains an element of the form
with a 
0,
then N = SL2(F).
Theorem 7.7.9.
Let F be any finite field with |F| > 3.
Then the projective special linear group PSL2(F)
is a simple group.
15.
Prove that there are no simple groups of order 200.
Solution
16. Sharpen Exercise 7.7.3 (b) of the text by showing that if G is a simple group that contains a subgroup of index n, where n > 2, then G can be embedded in the alternating group An. Solution
17. Prove that if G contains a nontrivial subgroup of index 3, then G is not simple. Solution
18. Prove that there are no simple groups of order 96. Solution
19. Prove that there are no simple groups of order 132. Solution
20. Prove that there are no simple groups of order 160. Solution
21. Prove that there are no simple groups of order 280. Solution
22. Prove that there are no simple groups of order 1452. Solution
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