Question

G is a finite group. We have shown that C(g) ≤ G for any g ∈...

G is a finite group. We have shown that C(g) ≤ G for any g ∈ G. Regarding the cosets of C(g):
1. The elements in the same coset all have something in common that distinguishes them
from the other cosets. Figure out what it is, state it clearly, and prove it.

2. Find a bijection between cl(g) and the set of cosets G/C(g) = { aC(g) | a ∈ G }. State
it clearly and prove that it is 1-1 and onto.
3. Use this bijection to explain why, in a finite group, |cl(g)| divides |G|.
4. If g ∈ Z(G), for any group G, what can you say about the sets cl(g) and C(g)?

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