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Groups : Isomorphism and Homomorphism

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Note:
S4 means symmetric group of degree 4
A4 means alternating group of degree 4
e is the identity

Is there a group homomorphism $:S4 -> A4, with
kernel $ = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}?

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No.
By the isomorphism theory, if there is a group homomorphism $: S4->A4, ...

Solution Summary

A group homomorphism is investigated.

$2.19
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Homomorphisms are examined.

EDIT: G(p) = {x in G : |x| = p^k for some k greater than or equal to 0}

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