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If all proper nontrivial subgroups of a nontrivial group are isomorphic to each other, must G...

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Basic Math

If all proper nontrivial subgroups of a nontrivial group areisomorphic to each other, must G be cyclic?

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Suppose that G has exactly two nontrivial proper subgroups H and K If K has a proper nontrivial subgroup then HK Thus K has exactly one proper nontrivial subgroup which means that K is cyclic of order p2 Hence G must be a pgroup of order p3 and any element xK will generate G If H and K do not have proper nontrivial subgroups H and K will be    See Answer
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