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Methods of derivation of semi-minor and pseudo-minor groups of $W$-symmetry. (Russian)

Authors: Lungu A. P.

Abstract

The left-quasihomomorphism $\mu$ of a group $G$ onto a subset $P^\prime$ of a group $P$ is defined by $\mu(g_1g_2)=\left[\mu(g_1)\right] \overset\leftharpoonup\to{\varphi_{g_2}}*\mu(g_2)=(p_1) \overset\leftharpoonup\to{\varphi_{g_2}}*p_2=p'_1*p_2=p_3$, where $g_1,g_2\in G; p_1,p_2,p_3\in P^\prime; \overset\leftharpoonup\to{\varphi_{g_2}}=\varphi(g_2), \varphi:G\rightarrow \Phi\leq \text{Aut} P$ a homomorphism The general properties of the map $\mu$ as well as the general structure of the $W^\prime$-semi-minor and $W^\prime$-pseudo-minor groups of $W$-symmetry $G^{(W)}$ (where: $G^{(W)}<P \bar \wr G, G=\{g\vert g^{(w)} \in G^{(W)}\}, W^\prime=\{w\vert g^{(w)}\in G^{(W)}\} \subset W=\overline{\prod}_{g_i\in G} P^{g_i}$ and $G^{(W)}\cap W^\prime=e$) are examined. One method of derivation of semi-minor and pseudo-minor groups of $W$-symmetry from permutation group $P$ and classical symmetry group $G$ is developed. This method is based on left-quasihomomorphisms of group $G$ onto a subset $W^\prime$ of a group $W$ (the action of automorphisms $\overset\leftharpoonup\to{\varphi_{g}}$ on the elements $w\in W^\prime$ is just the $g$-left-translation of their components).

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