# A group with a certain subgroup

Let $G$ be a group and has a subgroup $H$ such that $H$ contains all elements of $G$ of prime order. could we characterize such groups $G$ ?

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$H$ is a proper subgroup ? –  Amr Jan 22 '13 at 11:02
@ Amr! yes it is and $G$ is a torsion group. –  shankfei Jan 22 '13 at 11:11
I know one example --- the quaternion group (the nonabelian group of order 8 that's not the dihedral group). –  Gerry Myerson Jan 22 '13 at 11:19
thanks for your example! –  shankfei Jan 22 '13 at 11:54
There is definitely a classification for what $H$ looks like (assuming it is finite). –  user641 Jan 22 '13 at 12:40