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**Abstract Algebra Questions...**

I have two problems that I'm a little puzzled by, hopefully someone can shed some light.

1) Show that if H and K are subgroups of the group G, then H U K is closed under inverses.

2) Let G be a group, and let g ε G. Define the centralizer, Z(g) of g in G to be the subset

Z(g) = {x ε G | xg = gx}.

Prove that Z(g) is a subgroup of G.

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For problem 2 this is what I have but I am not sure if it is correct.

Since eg = ge for g in G, we know Z(g) is not the empty set.

-Take a in Z(g) and b in Z(g), and take any g in G, then we have...

(ab)g = a(bg) = a(gb) = (ag)b = (ga)b = g(ab). Thus ab is in Z(g).

- Take a in Z(g) and g in G. Then we know...

ag = ga

(a^-1* a )g = (a^-1 * g) a (multiplying both sides by a inverse)

e * g = a^-1 * g*a

g * a^-1 = a^-1 * g * (a * a^-1) ( multiplying again by a invese)

g * a^-1 = a^-1 * g

Thus a^-1 is in Z(g), so Z(g) is a subgroup of G.