Reference: Abstract Algebra, 3rd edition, I. N. Herstein, Prentice Hall International Edition.

Problems:

1. Let S be a set having an operation * which assigns an element a*b of S for any . Let us assume that the following two rules hold:

i) If a, b are any objects in S, then

ii) If a, b are any objects in S, then

Show that S can have at most one object.

II. Let S be the set of all integers . For a, b in S, define * by . Verify the following:

(i) unless .

(ii) in general. Under what conditions on a, b, c is ?

(iii) the integer 0 has the property that for every a in S.

(iv) For a in S, .

III) Let S consist oif f the two objects and . We define the operation * on S by subjecting and to the following conditions:

i) i

ii)

iii)

of verify by explicit calculations that if a, b, c are any elements of S (that is, a, b and c can be any of or then

i) is in S

ii)

iii)

iv) There is a particular a in S such that $la=latex a*b=b*a=b$ for all b in S.

,v) Given b in S, then where a is the parituclar element in part “iv” above.

Cheers,

Nalin Pithwa