28.9 HINTS AND ANSWERS
For Φ 4 ,K={(1),(123),(132)}is a subgroup.
Only Φ 1 is a homomorphism; Φ 4 fails because, for example, [(23)(13)]′ =
(23)′(13)′.
28.17 Recall that, for any pair of matricesPandQ,|PQ|=|P||Q|.Kis the set of all
matrices with unit determinant. The cosets ofKare the sets of matrices whose
determinants are equal;Kitself is the identity in the group of cosets.
28.19 (a) No, because the set is not closed, (b) yes, (c) yes, (d) yes.
28.21 Each subgroup contains the identity, a rotation byπ,andtworeflections.The
homomorphism is± 1 →I,±i→R^2 ,±j→mx,±k→mywith kernel{ 1 ,− 1 }.
28.23 There are 10 elements in all:I, rotationsRi(i=1,4) and reflectionsmj(j=1,5).
(a) There are five proper subgroups of order 2,{I,mj}and one proper subgroup
of order 5,{I, R,R^2 ,R^3 ,R^4 }.
(b) Four conjugacy classes,{I},{R, R^4 },{R^2 ,R^3 },{m 1 ,m 2 ,m 3 ,m 4 ,m 5 }.