MathDB

Problem 3

Part of 2019 LIMIT Category C

Problems(2)

statements about subgroup

Source: LIMIT 2019 CCS1 P3

4/28/2021
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}
group theoryabstract algebra
convergence of series

Source: LIMIT 2019 CCS2 P3

4/28/2021
Which of the following series are convergent? <spanclass=′latex−bold′>(A)</span> ∑n=1∞2n2+35n3+1<span class='latex-bold'>(A)</span>~\sum_{n=1}^\infty\sqrt{\frac{2n^2+3}{5n^3+1}} <spanclass=′latex−bold′>(B)</span> ∑n=1∞(n+1)nnn+3/2<span class='latex-bold'>(B)</span>~\sum_{n=1}^\infty\frac{(n+1)^n}{n^{n+3/2}} <spanclass=′latex−bold′>(C)</span> ∑n=1∞n2x(1−x2)n<span class='latex-bold'>(C)</span>~\sum_{n=1}^\infty n^2x\left(1-x^2\right)^n <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
seriesalgebraSummation