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Advanced Topics

From the isomorphism theorems to conjugacy classes and symmetric groups.

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Let \(H\) be a subgroup of \(G\) and let \(K\) be a subgroup of \(H.\) Consider the following two statements:

I. If \(K\) is a normal subgroup of \(G,\) then \(K\) is a normal subgroup of \(H.\)

II. If \(K\) is a normal subgroup of \(H\) and \(H\) is a normal subgroup of \(G,\) then \(K\) is a normal subgroup of \(G.\)

Which of these is/are always true?

Let \(M_2({\mathbb Z})\) be the group of \(2\times 2\) matrices with integer entries, with the group law given by addition. Let \(N\) be the subgroup consisting of matrices with even entries.

What group is \(\frac{M_2({\mathbb Z})}{N}\) isomorphic to?

Suppose \(G\) is a group of order \(60\) and \(g\in G\) is a non-identity element satisfying \(g^5 = 1.\) Let \(c\) be the number of distinct conjugates of \(g\) in \(G.\) What is the largest possible value of \(c\)?

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