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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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