Give an example of a cyclic subgroup of G and an example of a non-cyclicsubgroup of G. [3 marks](c) Show that K = {ρ0, ρ180} is a subgroup of G

Example of a Cyclic Subgroup

1. Cyclic Subgroup Example: Let $G = \mathbb{Z}_6$ (the integers modulo 6 under addition).
   • A cyclic subgroup of $G$ is generated by the element $2$: $\langle 2 \rangle = \{0, 2, 4\}$ This is a cyclic subgroup since every element can be expressed as $2k \mod 6$ for $k = 0, 1, 2$.

Example of a Non-Cyclic Subgroup

2. Non-Cyclic Subgroup Example: Consider $G = \mathbb{Z}_4 \times \mathbb{Z}_2$.
   • The subgroup $H = \{(0, 0), (0, 1), (2, 0), (2, 1)\}$ is non-cyclic. This subgroup cannot be generated by any single element from $H$, as there is no single element whose repeated addition (with respect to the group operation) can yield all other elements in $H$.

Show that $K = \{\rho_0, \rho_{180}\}$ is a Subgroup of $G$

3. Step 1: Verify Closure

   • Calculate:
     • $\rho_0 \cdot \rho_0 = \rho_0$
     • $\rho_0 \cdot \rho_{180} = \rho_{180}$
     • $\rho_{180} \cdot \rho_{180} = \rho_0$ Since the results are in $K$, the set is closed under the group operation.

4. Step 2: Identity and Inverses

   • The identity element in $G$, $\rho_0$, is in $K$.
   • The inverse of $\rho_0$ is $\rho_0$, and the inverse of $\rho_{180}$ is also $\rho_{180}$ since $\rho_{180} \cdot \rho_{180} = \rho_0$ (this shows $\rho_{180}$ is its own inverse).

5. Conclusion

   • Since $K$ satisfies the criteria of closure, contains the identity element, and has inverses for each of its elements, we conclude that $K$ is a subgroup of $G$.

Final Answer

1. Cyclic Subgroup: $\langle 2 \rangle = \{0, 2, 4\}$ in $G = \mathbb{Z}_6$.
2. Non-Cyclic Subgroup: $H = \{(0, 0), (0, 1), (2, 0), (2, 1)\}$ in $G = \mathbb{Z}_4 \times \mathbb{Z}_2$.
3. $K = \{\rho_0, \rho_{180}\}$ is a subgroup of $G$.