An Alternating Sum Formula for Multiplicities of Lower Defect Groups

Let G be a finite group and let k be an algebraically closed field of prime characteristic p. Let 5 b e a block of kG, P a /^-subgroup of G and n a /(-element of G. Then the multiplicity of P as a lower defect group of B for the /^-section of n is defined [1, 2, 4] and is denoted by mB(P). We shall prove the following induction theorem for mB(P) in terms of the Brown complex A of chains of non-trivial /^-subgroups of G.