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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