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Found problems: 2

1987 Traian Lălescu, 2.3
Tags: group theory , centralizer , normalizer , abstract algebra
Prove that $ C_G\left( N_G(H) \right)\subset N_G\left( C_G(H) \right) , $ for any subgroup $ H $ of $ G, $ and characterize the groups $ G $ for which equality in this relation holds for all $ H\le G. $ [i]Here,[/i] $ C_G,N_G $ [i]are the centralizer, respectively, the normalizer of[/i] $ G. $

2007 Nicolae Păun, 3
Tags: real analysis , group theory , limit , permutation , group of permutations , centralizer
In the following exercise, $ C_G (e) $ denotes the centralizer of the element $ e $ in the group $ G. $ [b]a)[/b] Prove that $ \max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right| <\frac{n!}{2} , $ for any natural number $ n\ge 4. $ [b]b)[/b] Show that $ \lim_{n\to\infty} \left(\frac{1}{n!}\cdot\max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right|\right) =0. $ [i]Alexandru Cioba[/i]