PROBLEM 3 Let n € Z. Let G = * for all k, mЄZ and z, wЄ Z2. Show that G is a group. (Z/nZ) × Z2 as a set, with binary operation defined by ([k], z) ⋆ ([m], w) = ([k + zm], zw)

9. Find all homomorphic images of the octic group.

39. Let be the set of all matrices in that have the form for arbitrary real numbers , , and . Prove or disprove that is a group with respect to multiplication.

Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.

Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.

49. Find the multiplicative inverse of in the given group. a. b.

Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.

9. Find all elements in each of the following groups such that . under addition. under multiplication.

42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .

Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an isomorphism from G to G that is different from the one given in Example 5 of this section. Example 5 Consider G=1,i,1,i under multiplication and G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. In order to define a mapping :G4 that is an isomorphism, one requirement is that must map the identity element 1 of G to the identity element [ 0 ] of 4 (part a of Theorem 3.30). Thus (1)=[ 0 ]. Another requirement is that inverses must map onto inverses (part b of Theorem 3.30). That is, if we take (i)=[ 1 ] then (i1)=((i))1=[ 1 ] Or (i)=[ 3 ] The remaining elements 1 in G and [ 2 ] in 4 are their own inverses, so we take (1)=[ 2 ]. Thus the mapping :G4 defined by (1)=[ 0 ], (i)=[ 1 ], (1)=[ 2 ], (i)=[ 3 ]