Some Involutions which Generate the Finite Symmetric Group
Abstract
Let $S_{n}$ be the symmetric group on $X_{n}=\{1, \dots, n\}$, for $n\geq 2$. In this paper we state some properties of subsemigroups generated by two involutions (a permutation with degree $2$) $\alpha,\beta$ such that $\alpha\beta$ is an $n$-cycle, and then state some generating sets of $S_n$ consists of involutions.
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References
- Ay\i k, G., Ay\i k, H., Bugay, L. and Kelekci, O., Generating sets of finite singular transformation semigroups, Semigroup Forum, 86 (2013), 59--66.
- Bugay, L., Quasi-idempotent ranks of some permutation groups and transformation semigroups, Turk. J. Math., Accepted.
- Ganyushkin, O. and Mazorchuk, V., Classical Finite Transformation Semigroups, Springer-Verlag, London, 2009.
- Garba, G. U., Idempotents in partial transformation semigroups, Proc. Royal Soc. Edinburgh, 116A (1990), 359--366.
- Garba, G. U., On the idempotent ranks of certain semigroups of order-preserving transformations,. Portugal. Math., 51 (1994) 185--204.
- Garba, G. U. and Imam, A. T., Products of quasi-idempotents in finite symmetric inverse semigroups, Semigroup Forum, 92 (2016), 645--658.
- Howie, J. M., Idempotent generators in finite full transformation semigroups, Proc. Royal Soc. Edinburgh, 81A (1978), 317--323.
- Howie, J. M., Fundamentals of Semigroup Theory, New York, Oxford University Press, 1995.
- Isaacs I. M., Finite Group Theory, American Mathematical Society, Graduate Studies in Mathematics, Volume 92, United States of America, 2008.