# Fundamental Mathematics Sample Problems

May I please have step-by-step workings for all attached problems?

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#### Solution Preview

** Please see the attached file for the complete solution **

1) We wish to verify that:

(please see the attached file)

and show that:

(please see the attached file)

is a normal subgroup of (please see the attached file). We also wish to find a homomorphism (please see the attached file) such that:

(please see the attached file)

(please see the attached file) is the subgroup of all even permutations of (please see the attached file) Since (please see the attached file) has 24 permutations, half of which are even, (please see the attached file) consists of 12 permutations. Thus it suffices to check that all permutations in the above expression for (please see the attached file) are even. The identity is even since it consists of 0 transpositions. The last three elements are even since each of them is the product of two disjoint transpositions. Finally, the eight remaining elements are cyclic permutations of length 3, each of which is the product of two ...

#### Solution Summary

In this solution we solve several problems in abstract algebra, mainly in group theory.