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statements about subgroup

Source: LIMIT 2019 CCS1 P3

April 28, 2021
group theoryabstract algebra

Problem Statement

GG be a group and H≤GH\le G. Then which of the following are true? <spanclass=′latex−bold′>(A)</span> a∈G,aHa−1⊂H⇒aHa−1=H<span class='latex-bold'>(A)</span>~a\in G,aHa^{-1}\subset H\Rightarrow aHa^{-1}=H <spanclass=′latex−bold′>(B)</span> ∃G,H and H≤G with H≅G<span class='latex-bold'>(B)</span>~\exists G,H\text{ and }H\le G\text{ with }H\cong G <spanclass=′latex−bold′>(C)</span> All subgroups are normal, then G is abelian.<span class='latex-bold'>(C)</span>~\text{All subgroups are normal, then }G\text{ is abelian.} <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
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