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Given the octic group $G = \{e, \sigma, \sigma^2, \sigma^3, \beta, \gamma, \delta, t\}$.

  1. Find a subgroup of $G$ that has order $2$ and is a normal subgroup of $G$.

  2. Find a subgroup of $G$ that has order $2$ and is not a normal subgroup of $G$.

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    @Jessica Sorry for all the comments. If I understand your question correctly, the following hint may be helpful: consider the subgroup generated by s^2 and the subgroup generated by b.2010-10-27

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Each subgroup of order 2 is generated by an element of order 2, and vice versa. So listing all the elements of order 2 would be a good start.

It might also help you to show that a subgroup of order 2 is normal if and only if the element of order 2 that generates it is in the center of $G$.