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lmedin02
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Homework Statement
Prove that the group of order 175 is abelian.
Homework Equations
The Attempt at a Solution
|G|=175=527. Using the Sylow theorems it can be determined that G has only one Sylow 2-subgroup of order 25 called it H and only one Sylow 7-subgroups called it K. Thus, H and K are normal subgroups of G and G=H x K which is isomorphic to the direct product of H and K. Since |H|=52, then H is Abelian. Since K is of prime order then K is cyclic and therefore also Abelian.
I am not sure whether I can now conclude that G must be abelian since it is the external (or direct) product of abelian subgroups.