By Kondratiev A. S., Mazurov V. D.

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I∈Y Thus the syndrome is an abelian group homomorphism. Let C = Ker s be its group kernel s−1 {0}. Note that |C| = |A|k . For x = i∈X bi in A, define xδ = i∈X bi − i∈X bi and xε = i∈X bi . Then xδ ∈ C and xε ∈ T , with x = xδ + xε . Thus An = C + T . |T |, so T is a loop transversal to C in An . Moreover, δ : An → C; x → xδ and ε : An → T ; x → xε surject, indeed ε|T = 1T , so T is precisely the set of errors corrected by C. ) For the alphabet Z/2Z = {0, 1}, consider the set (Z/2Z)4 = {0 = 0000, 1 = 0001, 2 = 0010, .

1. 8, ready for their application to the module theory of Chapter 10. 1 Combinatorial multiplication groups Let q be an element of a set (Q, ∗) equipped with a binary multiplication. The right multiplication RQ (q) or R∗ (q) is defined as the map R(q) : Q → Q; x → x ∗ q . 1) 35 36 An Introduction to Quasigroups and Their Representations The left multiplication LQ (q) or L∗ (q) is defined as the map L(q) : Q → Q; x → q ∗ x . 2) If (Q, ∗) is a quasigroup, then the right and left multiplications are elements of the group Q!

Qn ) ⇒ qEQ (p1 , . . , pm ) = qFQ (q1 , . . , qn ) ⇒ wE (q, p1 , . . , pm ) = wF (q, q1 , . . , qn ) ⇒ wE (q V , pV1 , . . , pVm ) = wF (q V , q1V , . . , qnV ) ⇒ EQV (pV1 , . . , pVm ) = FQV (q1V , . . , qnV ). 11) slightly, one obtains a combinatorial multiplication group functor Mlt from the category of surjective quasigroup homomorphisms to the category of group epimorphisms, taking a morphism f : P → Q to Mlt f : Mlt P → Mlt Q; EP (p1 , . . , pm ) → EQ (p1 f, . . , pm f ). 12) fails.

### 2-Signalizers of Finite Simple Groups by Kondratiev A. S., Mazurov V. D.

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