We are working on the proof of showing G (the group of rigid motions of a regular dodecahedron) is isomorphic to the alternating group A_5.
Lemma: Let H be a normal subgroup of a finite group G, and let x be an element of G. If o(x) and [G:H] are relatively prime, then x is in H.
Theorem: Any 60 element group having 24 el
Show that all automorphisms of a group G form a group under function composition.
Then show that the inner automorphisms of G, defined by f : G--->G so that
f(x) = (a^(-1))(x)(a), form a normal subgroup of the group of all automorphisms.
For the first part, I can see that we need to show that f(g(x)) = g(f(x)) for x in
Show that if H is any group then there is a group G that contains H as a normal subgroup with the property that for every automorphism f of H there is an element g of G such that the conjugation by g when restricted to H is the given automorphism f, i.e every automorphism of H obtained as an inner automorphism of G restricted to
Suppose F, E are fields and F is a subring of E. Prove that the set of ring isomorphisms Q:E-->E is a group Aut(E) under composition *, and that the set Aut(E/F) of isomorphisms Q in G, with Q(f)=f for all f in F is a subgroup of Aut(E).
If E is a field and H is a subgroup of Aut(E), show that the set E^H of elements of E t
Let C(a) be the conjugacy class in G containing a. Show that for a group G, if a <- G and f: G --> G is an automorphism, then b <- C(a) if and only if f(b) <- C(f(a)). Conclude that Aut G acts on the set of conjugacy classes of G.
Let G be a group, not necessarily finite, and let H be subgroup G.
(a) Prove that U = intersection of all x in G xHx^-1 is the largest
normal subgroup of G contained in H.
(b) Show that no proper subgroup H of A_5 contains six distinct Sylow
I need a detailed rigorous proof of this to study please.