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2-groups admit a difference bijection char(G character group character on G character sum character theoretic approach character theory characters of G class of characters consider construction cosets of H cyclic group defined due to Turyn element of G elementary abelian group equivalence class G and G g in G G/Ker(x group algebra group G group of order group of rank H into H Hadamard difference set homomorphism implies induced character induction integers iQ 0 otherwise isomorphic K-matrix in G K-matrix structure satisfying Ker(x|H kernel Lemma matrix associated multiplier need to check nonidentity element nonprincipal character O(mod odd first component odd powers order 22d+2 paper parameters preimage Proof quotient group rank d+1 rank(G restricted to H result root of unity satisfying properties set in G structure in G subgroup H subset Suppose T(y+jz techniques Theorem 4.4 three properties v,k,A)-difference set xr(sg yes Cor yes Thm