Abstract Algebra Answers

Q. (i) what is P-group?
(ii) Define order of P-group.

Q.Prove that two conjugate groups have same order.

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?