Complete mappings and incremental ratio in double-loops. (Italian. English summary) Zbl 0715.20045

Saeli, Donato

Riv. Mat. Univ. Parma, IV. Ser. 15, 111-117 (1989).

Let \((K,+,.)\) be a double loop, i.e. \((K,+)\) and \((K^*,.)\) are loops, where \(K^*\) is \(K\setminus \{0\}\), the neutral element in \((K,+)\). For a bijection f: \(K\to K\) and a fixed element \(a\in K\), the author defines the incremental ratio \(f_ a(x)=(f(x)-f(a))(x-a)^{-1}\), where -y denotes the inverse w.r.t. \(``+''\), and \(y^{-1}\) denotes the inverse w.r.t. “.”. Denote \(C(f)=\{a\in K|\) \(f_ a: K\setminus \{a\}\to K^*\) is a bijection\(\}\). The main results in the paper are: Theorem 1. If \((K,+,.)\) is a double loop satisfying the conditions that \((K,+)\) has the right inverse property and there exists a bijection f: \(K\to K\) with C(f)\(\neq \emptyset\), then there exists a bijection g of K such that \(0\in C(g)\) and, equivalently, \((K^*,.)\) has complete mappings. (A complete mapping of \((K^*,.)\) is a bijection \(\theta\) : \(K^*\to K^*\) such that \(\eta\) : \(x\to x\cdot \theta (x)\) is also a bijection of \(K^*.)\) Theorem 2. If \((K,+,.)\) is a double loop with the right inverse property for \((K,+)\), having \(4k+3\) elements and \((K^*,.)\) has a subloop with \(2k+1\) elements, then \(C(f)=\emptyset\) for a bijection f of K. Theorem 3. An associative nearfield \((N,+,.)\) is a double loop if and only if it is finite and the group \((N^*,.)\) has only cyclic Sylow 2-subgroups. The author proves that SL(2,q), for \(q=2^ h\) and \(h\geq 2\), and for \(q\in \{5,7,11\}\), has complete mappings.

Reviewer: M.Ştefănescu

MSC:

20N05	Loops, quasigroups
20D60	Arithmetic and combinatorial problems involving abstract finite groups
12K05	Near-fields

Keywords:

double loop; incremental ratio; right inverse property; complete mappings; associative nearfield; cyclic Sylow 2-subgroups