MATH 451 FIRST MID-TERM

NAME: John Q. Public

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2 MATH 451 FIRST MID-TERM

Question 1. Let Hbe a nonempty subset of the group G. Prove that His a

subgroup of Giff a b−1∈Hfor all a,b∈H.

First suppose that His a subgroup of G. Then His closed under multiplication

and taking inverses. Hence if a,b∈H, then b−1∈Hand so ab−1∈H.

Next suppose that ∅ 6=H⊆Gis such that ab−1∈Hfor all a,b∈H. Since

H6=∅, there exists an element a∈Hand hence 1 = aa−1∈H. It follows that if

a∈H, then a−1= 1 a−1∈H. Finally suppose that a,b∈H. Then b−1∈Hand

so ab =a(b−1)−1∈H. Thus His a subgroup of G.

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Mathematics

University:
Acharya Nagarjuna University

Subject:
Algebra

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23/02/2013