Menu Close

Let-H-be-a-non-empty-subset-of-a-group-G-prove-that-the-follow-ing-are-equivalent-1-H-is-a-subgroup-of-G-2-for-a-b-H-ab-1-H-3-for-a-b-ab-H-4-for-a-H-a-1-H-Hint-prove-1-2




Question Number 192077 by Mastermind last updated on 07/May/23
Let H be a non−empty subset of  a group G, prove that the follow−  ing are equivalent  1) H is a subgroup of G  2) for a,b ∈ H, ab^(−1)  ∈ H  3) for a,b ∈ ab ∈ H  4) for a ∈ H, a^(−1)  ∈ H    Hint: prove 1)→2)→3)→4)→1)    Help!!!
LetHbeanonemptysubsetofagroupG,provethatthefollowingareequivalent1)HisasubgroupofG2)fora,bH,ab1H3)fora,babH4)foraH,a1HHint:prove1)2)3)4)1)Help!!!
Answered by AST last updated on 07/May/23
H is a subgroup of G⇒H is a group with elements  from G.  ⇒ab∈H for a,b∈H  Since b∈H,b^(−1) ∈H  Hence,since a, b^(−1)  ∈H, ab^(−1) ∈H    ⇒ 1)⇒3)⇒2)⇒4)
HisasubgroupofGHisagroupwithelementsfromG.abHfora,bHSincebH,b1HHence,sincea,b1H,ab1H1)3)2)4)
Commented by Mastermind last updated on 07/May/23
Thank you so much sir
Thankyousomuchsir

Leave a Reply

Your email address will not be published. Required fields are marked *