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

I'm sorry, but you haven't provided enough information about the group G for me to give a specific example. However, I can give a general explanation.

(a) An example of a cyclic subgroup of a group G:

If G is the group of integers under addition, then an example of a cyclic subgroup is the set of even integers. This is because every even integer can be generated by repeatedly adding the generator of the subgroup, which is 2.

(b) An example of a non-cyclic subgroup of a group G:

If G is the group of all permutations of three elements, then an example of a non-cyclic subgroup is the set of all permutations that do not change the first element. This subgroup is not cyclic because there is no single permutation that can generate all others through repeated application.

(c) To show that K = {ρ0, ρ180} is a subgroup of G, we need to show that it satisfies the three subgroup tests:

1. Closure: If you combine any two elements in K, the result is still in K. Since ρ0 and ρ180 are inverses, their product is the identity, which is in K. The product of an element with itself is also in K because ρ0ρ0 = ρ0 and ρ180ρ180 = ρ0.

2. Identity: The identity element of G, ρ0, is in K.

3. Inverses: The inverse of ρ0 is ρ0, which is in K. The inverse of ρ180 is ρ180, which is also in K.

Therefore, K is a subgroup of G.