# Problems in Group Theory

See the attached file.

Suppose that G is a finite group and H is a normal subgroup of G.

(i) Explain briefly how the quotient group G/H is defined. [You should explain what the elements of G/H are, how the multiplication is defined, what the identity element is and how inverses are defined. You do not need to prove that the multiplication is well-defined.]

Show that the map

?:G?G/H

?(g)=gH

is a homomorphism. What is the kernel of ??

(ii) Hence show that if ?:G/H?GL(n,C) is a representation of the quotient group G/H then the map ? ?:G?GL(n,C) defined by

? ?(g)=?(gH)

(for each g?G) is a representation of G.

(iii) Now let G=A_4, the alternating group of even permutations of {1,2,3,4}. Let

H={e,(12)(34),(13)(24),(14)(23)}

Show that H?G. To which well-known group is the quotient group G/H isomorphic?

(iv) For G and H as in (iii) above, write down (without proof) three degree one representations of G/H.

(v) Hence describe three degree 1 (and hence irreducible) representations of G=A_4. Write down the characters of the representations you have found on representatives of the four conjugacy classes of G.

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#### Solution Summary

The solution solves several problems in group theory involving such topics as group representations, alternating groups, permutation groups, group homomorphisms, conjugacy classes, normal subgroups, general linear groups, and characters.