Problems Involving Free Groups and Free Products
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a. Show that in the free product, the center Z (G_1 x G_2) is trivial if |G_1| > 1 and |G_2| > 1.
b. Determine the elements of finite order in G_1 x G_2.
c. Show that the free group on a set X has no elements of finite order ( other than the identity).
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Solution Summary
We solve some group theory problems involving free groups and free products.
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For (a) and (b), we let G * H denote the free product of the groups G and H.
a. Show that Z = Z(G * H) is trivial if |G| > 1 and |H| > 1.
Proof: We have
Z = {z in G * H | forall t in G * H: zt = tz}.
Assume |G| > 1 and |H| > 1. Let g be a nontrivial element of G and h be a nontrivial element of H. Assume Z is nontrivial. Then there exists an element z of Z - ...
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