# group theory

1. Prove that the subgroup of A4 generated by any element of order 2 and and any element of order 3 is all of A4.

2. prove that if x and y are distinct 3-cycles in S4 with x != y^-1 (x not equal to the inverse of y), then the subgroup of S4 generated by x and y is all of A4.

Thanks. Any help is appreciated.

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#### Solution Preview

I need to use the following two Lemmas which are simple facts for group theory.

Lemma1: Suppose is a group and is a subgroup of . If , then is a normal subgroup of .

Lemma2: has unique normal subgroup .

Proof:

(a) Suppose are arbitrary ...

#### Solution Summary

This solution uses the following two Lemmas which are simple facts for group theory.

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