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Let (A, *) be a group. Let B be a subset of A such that 2|B| > a= b₁ |A|. Show that, for any a in A, -1 b₂ for some b₁ and b₂ in B. (Hint: Consider set C = (a b¹ | b = B}. ★

Let (A, *) be a group. Let B be a subset of A such that 2|B| > a= b₁ |A|. Show that, for any a in A, -1 b₂ for some b₁ and b₂ in B. (Hint: Consider set C = (a b¹ | b = B}. ★

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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each vertex of the graph 7. Prove that the order of any subgroup of a finite group divides the order of the group. 8. Let (A, v, ^,) be a Boolean Algebra. Show that (A, ) is an Abelian group, where is defined as ab=(a^ b)v (a^b). 9. Let (A, *) be a group. Let B be a subset of A such that 2|B| > |A|. Show that, for any a in A, a= b₁ b₂ for some b₁1 and b₂ in B. (Hint: Consider set C = (a b¹ | b = B).
Transcribed Image Text:each vertex of the graph 7. Prove that the order of any subgroup of a finite group divides the order of the group. 8. Let (A, v, ^,) be a Boolean Algebra. Show that (A, ) is an Abelian group, where is defined as ab=(a^ b)v (a^b). 9. Let (A, *) be a group. Let B be a subset of A such that 2|B| > |A|. Show that, for any a in A, a= b₁ b₂ for some b₁1 and b₂ in B. (Hint: Consider set C = (a b¹ | b = B).
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