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$H$ is a subgroup of $G$ and every coset of $H$ in $G$ is a subgroup of $G$.Then which of the following is true?
(A) $H=${$e$}
(B) $H=G$
(C) $G$ must have prime order.
(D) $H$ must have prime order.


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Can you rule out any of them? Can you see that any of them must hold? –  Tobias Kildetoft May 27 '13 at 16:55

2 Answers 2

up vote 0 down vote accepted

Hint:-
(B) is true. every subgroup of a group must contain the identity and any two distinct cosets are disjoint.

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In spite of being already given the final answer, think of the following: if we have a coset $\,gH\neq H\,$ , then it can not be that $\,1\in gH\,$ (why?) , thus...

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Dear Don, does $Q_8$ violet the problem? Thanks –  Babak S. May 27 '13 at 17:10
    
@BabakS. how could it? –  Tobias Kildetoft May 27 '13 at 17:20
    
I don't understand, @BabakS.: how and what does $\,Q_8\,$ violate ? –  DonAntonio May 27 '13 at 17:59
    
@DonAntonio: I felt that if I considered a right coset of, for example $H=\{\pm 1,\pm i\}$, then I would have another subgroup of $G=Q_8$ –  Babak S. May 27 '13 at 18:28
    
@DonAntonio: This means to me that I can consider $Q_8$ and a subgroup of it, say $H$, but none of the options are correct. Sometimes, I feel, I can't see the facts, however, I have been with them for times. Thanks for the time and (+1). –  Babak S. May 27 '13 at 18:34

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