Demonstration of the Fermat’s Little Theorem in Context of the Burnside Rings

We demonstrate proof of the Fermat’s little theorem in the context of the Burnside ring. 1 The Burnside ring of finite groups The Burnside ring B(G) of a finite group G is the Grothendieck ring of finite G-sets. It is generated as an algebra over Z by the isomorphism classes of finite (left) G-sets S, T, · · · , subject to the relations S − T = 0, if S ∼ = T, S + T − (S ∪ T) = 0, S · T − (S × T) = 0 Therefore, the elements of B(G) are the virtual G-sets; that is, the formal differences S − T of isomorphism classes of G-sets S, T; (see [1], [3]). For reader’s convenience, we start by writing down some of the notations used in this paper. Notation 1.1 Let H,K be subgroups of a finite group G. We say that H and K are G-conjugate and write H ∼ G K if g−1Hg = K for some g ∈ G. If g−1Hg ⊆ K for some g ∈ G, we write H G K, and say that H is G-subconjugate to K. We denote the set of G-invariant subset of a G-set S by FixG(S): = {s ∈ S | gs = s ∀ g ∈ G}; in particular, for K ≤ G, we have that FixH(G/K) = {gK | g ∈ G, g−1Hg ≤ K}. 2192 Kenneth K. Nwabueze The following results are well known (see [1], [3]). Proposition 1.1 Let H and K be subgroups of G, and let S be any G-set. Then (i) there exists a bijection HomG(G/H,G/K) ↔ FixH(S), (ii) there exists a bijection HomG(G/H,G/K) ↔ FixH(G/K), (iii) FixH(G/K) = ∅ unless