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Groups and subgroups

  • Thread starter xlalcciax
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  • #1
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1. Let G be a group containing subgroups H and K such that we can find an element h e H-K an an element k e K - H. Prove that h o k is not a subgroup of H U K. Deduce that H U K is not a subgroup of G.

I have proved that h o k is not in H U K but I dont know how to deduce that H U K is not a subgroup of G.
 

Answers and Replies

  • #2
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If [tex]h \in H \setminus K[/tex], [tex]k \in K \setminus H[/tex], and you have proved that [tex]hk \notin H \cup K[/tex], that shows that [tex]H \cup K[/tex] fails to satisfy one of the three basic properties required of a group. Which one?
 
  • #3
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If [tex]h \in H \setminus K[/tex], [tex]k \in K \setminus H[/tex], and you have proved that [tex]hk \notin H \cup K[/tex], that shows that [tex]H \cup K[/tex] fails to satisfy one of the three basic properties required of a group. Which one?
Inverse element?
 
  • #4
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No, guess again...
 
  • #5
SammyS
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If [tex]h \in H \setminus K[/tex], [tex]k \in K \setminus H[/tex], and you have proved that [tex]hk \notin H \cup K[/tex], that shows that [tex]H \cup K[/tex] fails to satisfy one of the three basic properties required of a group. Which one?
Inverse element?
Hi xlalcciax.

Are h, k ϵ H∪K ?
 

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