UDK 519.6
SHUNKOV GROUPS, SATURATED BY GROUPS L2(pn), U3(2n)
E. A. Pronina1, A. A. Shlepkin2*
1Krasnoyarsk State Аgrarian University 44 I, Stasova St., Krasnoyarsk, 660130, Russian Federation 2Siberian Federal University 79, Svobodny Av., Krasnoyarsk, 660041, Russian Federation *E-mail: ak_kgau@mail.ru
Investigated are Shunkov groups, saturated by groups (projective special linear group of degree 2 over finite fields), (projective special unitary group of degree 3 over fields of odd characteristics). Arbitrary group is called a Shunkov group, if every cross section by a finite subgroup of any pair of conjugate elements of Prime order generates a finite subgroup. Under periodic part group G is the subgroup generated by all elements of finite order of G, provided that it is periodic. Presented is a series of lemmas, in which we prove that: – G contains infinitely many elements of finite order; – In G there are finite subgroups K1 and K2, that and , but for no group of such that ; – Sylow 2-subgroup S, group G, locally finite and for any ; – All involution of S lies in – For any with the property it follows that ; – If V is a Sylow 2-subgroup of G and , then ; – All Sylow 2-subgroups of G are conjugate; – If and , then ; – Subgroup has a periodic part , where H is a locally periodic cyclic group without involutions; – The subgroup B is embeddable in locally finite simple subgroup L group G that is isomorphic U3(Q), where Q is a locally finite field of characteristic 2; – If the a for arbitrary nonunit element of H0, then has a periodic part N and , where t is an involution. Based on the above lemmas, we prove the theorem: the Shunkov group saturated by multiple groups of the form , has a periodic part , is isomorphic to either , or , for suitable locally of finite fields P and Q.
Keywords: group Shunkov, saturation, periodic part.
References
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Pronina Ekaterina Alekseevna – postgraduate student, Krasnoyarsk State Agrarian University. E-mail: katyushka_2707@mail.ru.
Shlepkin Alexey Anatolievich – Docent, Siberian Federal University. E-mail: ak_kgau@mail.ru.