Math787_HW1

# Math787_HW1 - G When is this map a homomorphism 4 Let A and...

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Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Homework I: Group Theory 1. Let f : G H be a bijective homomorphism of groups. Show that f - 1 is also a homomorphism. 2. Let G be a group of order 2 p for some odd prime p . Show that G has an odd number of elements of order 2. 3. Let G be a ﬁnite group of odd order. Show that the map x 7→ x 2 is a surjective map from G to
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Unformatted text preview: G . When is this map a homomorphism? 4. Let A and B be subgroups of G . When is A ∪ B a subgroup of G ? 5. Let G be a group having no proper subgroups. Show that G is cyclic, ﬁnite and of prime order. 6. Let S be a proper subgroup of G (i.e. S 6 = { e } and S 6 = G .) Show that the subgroup generated by the set G-S = G . 1...
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## This note was uploaded on 07/17/2008 for the course MATH 787 taught by Professor Joshua during the Summer '08 term at Ohio State.

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