One of the interesting classes of mixed groups ( i.e. groups that can contain both elements of finite order and elements of infinite order) is the class of Shunkov groups. The group 𝐺 is called Shunkov group if for any finite subgroup 𝐻 of 𝐺 in the quotient group 𝑁_G(𝐻) / 𝐻, any two conjugate elements of prime order generate a finite group. When studying the Shunkov group 𝐺, a situation often arises when it is necessary to move to the quotient group of the group 𝐺 by some of its normal subgroup 𝑁. In which cases is the resulting quotient group 𝐺/𝑁 again a Shunkov group? The paper gives a positive answer to this question, provided that the normal subgroup 𝑁 is locally finite and the orders of elements of the subgroup 𝑁 are mutually simple with the orders of elements of the quotient group 𝐺/𝑁. Let X be a set of groups. A group 𝐺 is saturated with groups from the set X if any finite subgroup of 𝐺 is contained in a subgroup of 𝐺 that is isomorphic to some group of X . If all elements of finite orders from the group 𝐺 are contained in a periodic subgroup of the group 𝐺, then it is called the periodic part of the group 𝐺 and is denoted by 𝑇(𝐺). It is proved that the Shunkov group saturated with finite linear and unitary groups of degree 3 over finite fields has a periodic part that is isomorphic to either a linear or unitary group of degree 3 on a suitable locally finite field.

Aleksei Shlepkin, Cand. Sci. (Phys.–Math.), Siberian Federal University, 79, Svobodny pr., Krasnoyarsk, 660041, Russian Federation, tel.: +7 (391) 291-28-64, e-mail: shlyopkin@mail.ru

Irina Sabodakh, Cand. Sci. (Phys.–Math.), Siberian Federal University, 79, Svobodny pr., Krasnoyarsk, 660041, Russian Federation, tel.: +7 (391) 291-28-64, e-mail: sabodakh@mail.ru

Shlepkin A.A., Sabodakh I.V. On Two Properties of Shunkov Groupals. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 35, pp. 103-119. (in Russian) https://doi.org/10.26516/1997-7670.2021.35.103

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