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Large sets of idempotent quasigroups with prescribed conjugate-invariant subgroup of S_3

Denote by $S_{3}$ the symmetric group of all permutations on threeelements of $\{1,2,3\}$.

For an ordered triple $B=(x_1,x_2,x_3)$ and a permutation $\sigma\in S_{3}$, we define $B^\sigma=(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)})$,meaning that $\sigma$ acts on the column labels. If ${\cal B}$ is a collectionof ordered triples of $X$, define ${\cal B}^{\sigma}=\{B^{\sigma}: B\in {\calB}\}$. Let $\Omega=\{(X, Q_i): i=1,2,\ldots, v-2\}$ be a large sets ofidempotent quasigroups and $H$ be a given subgroup $H$ of $S_3$. If$\Omega^{h}=\Omega$ for any $h\in H$ where $\Omega^{h}=\{(X,Q_i^h):i=1,2,\ldots,v-2\}$, then $\Omega$ is said to be {\it $H$-conjugate invariant}.In this talk, we consider the existence of large sets of idempotent quasigroupswith prescribed conjugate-invariant subgroup $H$, where $H=(1), C_2, C_3$.

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