Alternating groups

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}.

Normal subgroups

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.

Automorphism

Show that innG is the normal subgroup of autG for any group G Note: innG = inner automorphism group of G aut G = automorphism group of G

Groups and Subgroups : Indicies

If K is a normal subgroup of G has index m, show that g^m belongs to K for all g belonging to G.

Subgoups : Indicies

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|

Groups : Symmetry

Note: G' means derived (commutator) subgroup of G and Sn is symmetric group of degree n Please find G' in each case (a) G is abelian (b) G = Sn

Groups : Isomorphism and Homomorphism

Note: S4 means symmetric group of degree 4 A4 means alternating group of degree 4 e is the identity Is there a group homomorphism $:S4 -> A4, with kernel $ = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}?

Groups : Isomorphism and Homomorphism

Show that a group G is simple if and only if every nontrivial group homomorphism G -> G1 is one-to-one.

Groups : Isomorphism and Homomorphism

Note: G =~ G1 means G is isomorphic to G1 If G/K =~ H, show that there exists an onto homomorphism $:G -> H with kernel $ = K

Rings and Units

Given r and s in a ring R, show that 1 + rs is a unit if and only if 1 + sr is a unit.