Given the octic group $G = \{e, \sigma, \sigma^2, \sigma^3, \beta, \gamma, \delta, t\}$.
Find a subgroup of $G$ that has order $2$ and is a normal subgroup of $G$.
Find a subgroup of $G$ that has order $2$ and is not a normal subgroup of $G$.
Given the octic group $G = \{e, \sigma, \sigma^2, \sigma^3, \beta, \gamma, \delta, t\}$.
Find a subgroup of $G$ that has order $2$ and is a normal subgroup of $G$.
Find a subgroup of $G$ that has order $2$ and is not a normal subgroup of $G$.
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 $G$.