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**Lemma 7.7.1.**
If n 3,
then every permutation in A_{n}
can be expressed as a product of 3-cycles.

**Theorem 7.7.2.**
The symmetric group S_{n} is not solvable for
n 5.

**Lemma 7.7.3.**
If n 4,
then no proper normal subgroup of A_{n} contains a 3-cycle.

**Theorem 7.7.4.**
The alternating group A_{n} 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 SL_{n}(F).

The group SL_{n}(F) modulo its center is called the
** projective special linear group**
and is denoted by PSL_{n}(F).

**Proposition 7.7.6.**
For any field F, the center of SL_{n}(F)
is the set of nonzero scalar matrices with determinant 1.

**Example.** 7.7.1.
PSL_{2}(F) S_{3}
if |F| = 2.

**Example.** 7.7.2.
PSL_{2}(F)
A_{4} if |F| = 3.

**Lemma 7.7.7.**
Let F be any field. Then SL_{2}(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 SL_{2}(F).
If N contains an element of the form
with a 0,
then N = SL_{2}(F).

**Theorem 7.7.9.**
Let F be any finite field with |F| > 3.
Then the projective special linear group PSL_{2}(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 A_{n}.
*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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