abstract algebra

Note: e is identity and A4 is the alternating group of degree 4. Let K = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. Show that K is the only normal subgroup of A4 apart from A4 and {e}. Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks.

abstract algebra

Let X be a nonempty subset of a group G. If G = and H is a subgroup of G, show that H is the normal subgroup of G if and only if x^-1Hx contained in H for all x belonging to X. ALSO show that is normal in G if and only if gXg^-1 contained in for all g belonging to G. Please be very spcefic and justify your answer so I can understand. The more thorough the better. Please only answer if willing to complete both parts. Any questions please ask. Thanks.

abstract algebra

Show that innG is the normal subgroup of autG for any group G Note: innG = inner automorphism group of G aut G = automorphism gourp of G Please be very spcefic and justify your answer so I can understand. The more thorough the better. Any questions please ask. Thanks.

factor groups

Please, if you are going to answer this question, include as much detail as you can so that I can follow what your doing. Thank you very much! If K is a normal subgroup of G has index m, show that g^m belongs to K for all g belonging to G

factor groups

note: C means set containment (not proper set containment), |G : K| means index of subgroup K in G, and G # K means K is a normal subgroup of G question: Let K C H C G be groups, where K # G and |G : K| is finite. Show that |G/K : H/K| is also finite and that |G/K : H/K|=|G : H| Please give me any comments on my posting if you do not want to do it.