Abstract Algebra Answers

Q. Prove that center of a group is itself a subgroup.

Q. prove that
(i) H={a+ib ЄC, a2+b2=1} is a subgroup of C, Where C is complex number.
(ii) H be a set of real number a+b√2 where a,b ϵ Q. Show that H be a subgroup of non-zero real no. under *.

Q. prove that a non-commutative group has at least six elements.

Q. prove that every group of prime order is cyclic and hence abelian.

Q. Is H=<Q', *> is a subgroup of G=<R, +>

Q. Prove that <z,+> is a group, H=<5z,+> is a subgroup of G=<Z,+>

Q. Do the non-zero positive rational no. form a group w.r.t multiplication?
Q.2: Do the even integer form a group w.r.t addition?

Q.1: Do the odd integer form a group w.r.t addition?
Q.2: Do the non-zero positive real number form a group w.r.t multiplication?

Q. Find index of subgroup generated by 3 in Z12.

Q. prove that H={a+ib C, a2+b2=1} is a subgroup of C. Where C is complex number. H be set of real number a+b where a,b Є ф. Show that H be a subgroup of non-zero real no. under multiplication.