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# Problems in Group Theory

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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.