Direct sum for Abelian groups

please see attached If B is an Abelian group and ... a family of homomorphisms, prove that there is a unique homomorphism...

Coproduct

please see attached Prove that the free product...is a coproduct in the category of groups.

Nilpotent group

please see attached Let N be a nontrivial normal subgroup of a nilpotent group G...

Inner automorphism

please see attached If G is a group and x is in G, define the inner automorphism f_x by setting...

Nilpotent groups and maximal subgroups

please see attached Let G be a finite group in which every maximal subgroup is normal...

Nilpotent proofs

please see attached Show that subgroups and homomorphic images of nilpotent groups are nilpotent.

Group Theory Proof

a) prove that if G is a finite group and a is an element of G then for some positive m , a^m is equal to the identity of G. (Use the Pigeon hole principle) b) Prove that if G is a finite group, H subset of G that is closed with respect to the operation of G, Then every element of H has its inverse in H. thanks.

Category of nilpotent groups

Prove that there cannot be a nilpotent group N generated by two elements with the property that every nilpotent group generated by two elements is a homomorphic image of N (i.e.: free objects do not always exist in the category of nilpotent groups).

Let G be a group with X a subset of G and let A be the normal subgroup generated by X

(i.e.: A = the intersection {N a normal subgroup of G: X subset of N} A=bigcap {Nlefttriangle G:Xsubseteq N}-this is the tex for what was written on the study guide.) Let Y = {gxg^-1|x is X, g in G}. Show that A= I need a rigorous proof with explanations so that I can study and understand please. I have an exam ...continues

Presentations

Show that the Klein 4 group V has presentation < a,b|a^2=b^2=(ab)^2=1 > I need a rigorous proof with explanations so that I can study and understand please. I have an exam on Thursday.