Nonempty Subset - Abstract Algebra - Exam, Exams for Algebra. Acharya Nagarjuna University


Description: This is the Exam of Abstract Algebra which includes Weighted Equally, Group Theory, Vector Spaces, Linear Algebra, Unique Factoriztaion, Normal Subgroup etc. Key important points are: Nonempty Subset, Group, Subgroup, Multiplication, Element, Isomorphic, Symmetric Group, Alternating Group, Contradiction, Subgroup
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NAME: John Q. Public
Question Marks
1 12
2 25
3 28
4 25
5 10
Question 1. Let Hbe a nonempty subset of the group G. Prove that His a
subgroup of Giff a b1Hfor all a,bH.
First suppose that His a subgroup of G. Then His closed under multiplication
and taking inverses. Hence if a,bH, then b1Hand so ab1H.
Next suppose that ∅ 6=HGis such that ab1Hfor all a,bH. Since
H6=, there exists an element aHand hence 1 = aa1H. It follows that if
aH, then a1= 1 a1H. Finally suppose that a,bH. Then b1Hand
so ab =a(b1)1H. Thus His a subgroup of G.
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