Chapter 3.3, Problem 1E

(a)

To determine

Whether the set $\left\{e,\sigma \right\}$ is or is not a subgroup of $S(A)=\left\{e,\rho ,{\rho }^2,\sigma ,\gamma ,\delta \right\}$ by constructing a multiplication table for the given subset

(b)

To determine

Whether the set $\left\{e,\delta \right\}$ is or is not a subgroup of $S(A)=\left\{e,\rho ,{\rho }^2,\sigma ,\gamma ,\delta \right\}$ by constructing a multiplication table for the given subset

(c)

To determine

Whether the set $\left\{e,\rho \right\}$ is or is not a subgroup of $S(A)=\left\{e,\rho ,{\rho }^2,\sigma ,\gamma ,\delta \right\}$ by constructing a multiplication table for each subset

(d)

To determine

Whether the set $\left\{e,{\rho }^2\right\}$ is or is not a subgroup of $S(A)=\left\{e,\rho ,{\rho }^2,\sigma ,\gamma ,\delta \right\}$ by constructing a multiplication table for each subset

(e)

To determine

Whether the set $\left\{e,\rho ,{\rho }^2\right\}$ is or is not a subgroup of $S(A)=\left\{e,\rho ,{\rho }^2,\sigma ,\gamma ,\delta \right\}$ by constructing a multiplication table for each subset.

(f)

To determine

Whether the set $\left\{e,\rho ,\sigma \right\}$ is or is not a subgroup of $S(A)=\left\{e,\rho ,{\rho }^2,\sigma ,\gamma ,\delta \right\}$ by constructing a multiplication table for each subset

(g)

To determine

Whether the set $\left\{e,\sigma ,\gamma \right\}$ is or is not a subgroup of $S(A)=\left\{e,\rho ,{\rho }^2,\sigma ,\gamma ,\delta \right\}$ by constructing a multiplication table for each subset

(h)

To determine

Whether the set $\left\{e,\sigma ,\gamma ,\delta \right\}$ is or is not a subgroup of $S(A)=\left\{e,\rho ,{\rho }^2,\sigma ,\gamma ,\delta \right\}$ by constructing a multiplication table for each subset.