On finite irreducible subgroups of the group $GL(q,R_p)$

A. Kirilyuk

All irreducible $p$-subgroups of the groups $GL(\!q,R_p\!)$ ($q$ is a prime) are described up to conjugation in the case $R_p$ is a ring of integers of the finite extension $F_p$ of the rational $p$-adic field $\mathbb{Q}_p$ and $(F_p : \mathbb{Q}_p)\leq 2$.

1. Suprunenko~D.~A. The minimal irreducible solvable subgroups of the prime degree, Trudy Mosk. Matem. Ob., 29 (1973), 223--334 (in Russian).

2. Yuferev~V.~P. The classifikation of the minimal irreducible linear groups of the prime degree, Izv. AN BSSR. Ser. Phis.-Mat. Nauk, 2 (1974), 5--10 (in Russian).

3. Kirilyuk~A.A. The minimal irreducible solvable subgroups of the group $GL(2,R_p)$, Materialy XXXII Naych. Conf. Uzhgorod Univ., Uzhgorod, Dep., No.~2079--79, (1978), 166--198 (in Russian).

4. Kirilyuk~A.~A. The irreducible solvable $p$-subgroups of the group $GL(3,R_p)$, Nauk. Visnik Uzhgorod. Univ. Ser. Mat. i Inform., 7 (2002) 49--57.

5. Gudivok~P.~M. and Kirilyuk~A.~A. On minimal irreducible subgroups of the general linear group over the ring of $P$-adic integer, Nauk. Visnik Uzhgorod. Univ. Ser. Mat. i Inform., 7 (2002), 37--437.

6. Suprunenko~D.~A. Linear $p$-groups, Dokl. AN BSSR, 4 (1960), No.~6, 233--235 (in Russian).

7. Vol'vathov~R.~T. $p$-subgroups of Sylow of the general linear group, Izv. AN USSR, math., 27 (1963), No.5, 1031--1954 (in Russian).

8. Leedham-Green~C.~R., Plesken~W. Some remarks of Sylow subgroups linear groups, Math. Z., 191, (1980), 529--535.

9. Konyuch~V.~S. On linear $p$-groups, Izv. AN BSSR, ser. fiz.-math., (1987), 3--8 (in Russian).

10. Gudivok~P.~M. and Kirilyuk~A.~A. Sylow $p$-subgroups of the general linear group over discrete valuation rings, Dokl. AN Ukr. SSR, ser.~A, (1979), № 5, 326--329 (in Russian).

11. Gudivok~P.~M., On Sylow $p$-subgroups of the general linear group over complete discrete valuation rings, Ukr. math. j. 43 (1991), No.~7--8, 918--924.

12. Vaschuk~F.~G. and Gudivok~P.~M. On integral $p$-adic representation of the finite abelian $p$-groups, Dop. AN Ukr. SSR, ser.~A, N.~1 (1986), 3--6.

13. Gudivok~P.~M. and Rud'ko~V.~P. On Sylow subgroups of the general linear group over integral domains, Dop. NAN Ukraine, (1995), No.8, 5--7.

14. Gudivok~P.~M., Rud'ko~V.~P., Yurthcenko~N.~V. On Sylow $p$-subgroups of the general linear groups over principal integral domains, Nauk. visnik Uzhgorod. un. Ser. math. and inform., 6, (2001), 31--46.

15. Kirilyuk~A.~A. Irreducible Sylow $p$-subgroups of $GL(3,R_p)$, Nauk. visnik Uzhgorod. un. Ser. math. and inform., 7 (2002), 49--57.

16. Sapujak~T.~M. $p$-subgroups of the general linear group over $P$-adic qudratic rings, Uzhgorod, Dep., No.~1778--Uzhg.86, (1986), 33~pp (in Russian).

17. Hasse~H. Zahlentheorie, Berlin, 1963.

18. Kirilyuk~A.~A. Finite subgroups of the group $GL(2,\mathbb{Z}_2[i])$, Uzhgorod, Dep. N.~60575, (1976), 106--113 (in Russian).

19. Kirilyuk~A.~A. Finite subgroups of the group $GL(3,\mathbb{Z}_2)$, Theor. and applied pr. of dif. equations and algebra, Kiev, (1978), 114--119.

	On finite irreducible subgroups of the group $GL(q,R_p)$
Author	A. Kirilyuk algebra@tn.uz.ua Faculty of Mathematics, Uzhgorod National University, Uzhgorod, Universitetska, 14
Abstract	All irreducible $p$-subgroups of the groups $GL(\!q,R_p\!)$ ($q$ is a prime) are described up to conjugation in the case $R_p$ is a ring of integers of the finite extension $F_p$ of the rational $p$-adic field $\mathbb{Q}_p$ and $(F_p : \mathbb{Q}_p)\leq 2$.
Keywords	irreducible subgroup, ring of integer, rational p-adic field
DOI	doi:10.30970/ms.22.2.152-158
Reference	1. Suprunenko~D.~A. The minimal irreducible solvable subgroups of the prime degree, Trudy Mosk. Matem. Ob., 29 (1973), 223--334 (in Russian). 2. Yuferev~V.~P. The classifikation of the minimal irreducible linear groups of the prime degree, Izv. AN BSSR. Ser. Phis.-Mat. Nauk, 2 (1974), 5--10 (in Russian). 3. Kirilyuk~A.A. The minimal irreducible solvable subgroups of the group $GL(2,R_p)$, Materialy XXXII Naych. Conf. Uzhgorod Univ., Uzhgorod, Dep., No.~2079--79, (1978), 166--198 (in Russian). 4. Kirilyuk~A.~A. The irreducible solvable $p$-subgroups of the group $GL(3,R_p)$, Nauk. Visnik Uzhgorod. Univ. Ser. Mat. i Inform., 7 (2002) 49--57. 5. Gudivok~P.~M. and Kirilyuk~A.~A. On minimal irreducible subgroups of the general linear group over the ring of $P$-adic integer, Nauk. Visnik Uzhgorod. Univ. Ser. Mat. i Inform., 7 (2002), 37--437. 6. Suprunenko~D.~A. Linear $p$-groups, Dokl. AN BSSR, 4 (1960), No.~6, 233--235 (in Russian). 7. Vol'vathov~R.~T. $p$-subgroups of Sylow of the general linear group, Izv. AN USSR, math., 27 (1963), No.5, 1031--1954 (in Russian). 8. Leedham-Green~C.~R., Plesken~W. Some remarks of Sylow subgroups linear groups, Math. Z., 191, (1980), 529--535. 9. Konyuch~V.~S. On linear $p$-groups, Izv. AN BSSR, ser. fiz.-math., (1987), 3--8 (in Russian). 10. Gudivok~P.~M. and Kirilyuk~A.~A. Sylow $p$-subgroups of the general linear group over discrete valuation rings, Dokl. AN Ukr. SSR, ser.~A, (1979), № 5, 326--329 (in Russian). 11. Gudivok~P.~M., On Sylow $p$-subgroups of the general linear group over complete discrete valuation rings, Ukr. math. j. 43 (1991), No.~7--8, 918--924. 12. Vaschuk~F.~G. and Gudivok~P.~M. On integral $p$-adic representation of the finite abelian $p$-groups, Dop. AN Ukr. SSR, ser.~A, N.~1 (1986), 3--6. 13. Gudivok~P.~M. and Rud'ko~V.~P. On Sylow subgroups of the general linear group over integral domains, Dop. NAN Ukraine, (1995), No.8, 5--7. 14. Gudivok~P.~M., Rud'ko~V.~P., Yurthcenko~N.~V. On Sylow $p$-subgroups of the general linear groups over principal integral domains, Nauk. visnik Uzhgorod. un. Ser. math. and inform., 6, (2001), 31--46. 15. Kirilyuk~A.~A. Irreducible Sylow $p$-subgroups of $GL(3,R_p)$, Nauk. visnik Uzhgorod. un. Ser. math. and inform., 7 (2002), 49--57. 16. Sapujak~T.~M. $p$-subgroups of the general linear group over $P$-adic qudratic rings, Uzhgorod, Dep., No.~1778--Uzhg.86, (1986), 33~pp (in Russian). 17. Hasse~H. Zahlentheorie, Berlin, 1963. 18. Kirilyuk~A.~A. Finite subgroups of the group $GL(2,\mathbb{Z}_2[i])$, Uzhgorod, Dep. N.~60575, (1976), 106--113 (in Russian). 19. Kirilyuk~A.~A. Finite subgroups of the group $GL(3,\mathbb{Z}_2)$, Theor. and applied pr. of dif. equations and algebra, Kiev, (1978), 114--119.
Pages	152-158
Volume	22
Issue	2
Year	2004
Journal	Matematychni Studii
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