Symmetry And Group

Symmetry of positive solutions of an almost-critical problem by Castorina D., Pacella F.

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By Castorina D., Pacella F.

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1. In other words, the orbit set Q/N carries a quasigroup structure with xN · yN = (xy)N for x, y in Q. Finally, it is worth remarking that N → N is a closure operator on the set of normal subgroups of the combinatorial multiplication group Mlt Q of the quasigroup Q. 4 Inner multiplication groups of piques For an element e of a (nonempty) quasigroup Q with combinatorial multiplication group G, let Ge denote the stabilizer {g ∈ G | eg = e} of e in G. Note that for each element g of G, the stabilizer Geg is the conjugate Gge = g −1 Ge g of Ge by g.

Two words are said to be σ-equivalent if they are related by a (possibly empty) sequence of such replacements. Note that if a word w contains r letters from µS3 , then it has 2r σ-equivalent forms (Exercise 31). A word w from W is said to be primary if it does not include the letters µσ , µστ , µτ σ (the opposites of the respective basic quasigroup operations ·, \, /). Each σ-equivalence class has a unique primary representative. The normal form is chosen as the primary representative of its σ-equivalence class.

Closure under right division follows by symmetry. Thus V is a subquasigroup of Q × Q. Conversely, suppose V is a congruence on Q. For q in Q and (x, y) in V , one has (x, y)R(q) = (xR(q), yR(q)) = (xq, yq) = (x, y)(q, q) ∈ V . and similarly (x, y)R(q)−1 = (x, y)/(q, q) ∈ V , (x, y)L(q) = (q, q)(x, y) ∈ V , (x, y)L(q)−1 = (q, q)\(x, y) ∈ V . Thus V is an invariant subset of the G-set Q × Q. Recall that the action of a group H on a set X is said to be primitive if it is transitive, and the only H-congruences on X are the trivial congruence X and the improper congruence X 2 .

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