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Units in the modular group algebra of a finite Abelian p-group. (English) Zbl 0543.20008

The author presents two minimal generating sets for the group V of normalized units in \({\mathbb{F}}_ pA\), A an abelian p-group. One is given as follows: Let A have ”basis” \(x_ i\) of order \(q_ i\), \(1\leq i\leq d\) and \(let\)
D\(=\{\delta =(\delta_ 1,...,\delta_ d): \delta_ i\in {\mathbb{N}}_ 0\), \(0\leq \delta_ i<q_ i\), \(p| \delta_ j\) for some \(j\}\).
For \(\delta \in D\) put \(P(\delta)=\prod_{1\leq j\leq d}(x_ j- 1)^{\delta_ i}\). Then \(\{1+P(\delta): \delta \in D\}\) is a basis for V. A similar basis is given in terms of the socle of G. Also the invariants and the rank of V are determined. Finally the author proves the interesting result: Let A be a finite abelian p-group. Then a subgroup U of \(V({\mathbb{F}}_ pA)\) which is linearly independent as a subset of \({\mathbb{F}}_ pA\) is isomorphic to a subgroup of A. The analogous result is also true for the non-abelian group of order \(p^ 3\), but not for the quaternion group of order 16.
Reviewer: K.W.Roggenkamp

MSC:

20C20 Modular representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16U60 Units, groups of units (associative rings and algebras)
16S34 Group rings
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[1] Bass, H., Algebraic \(K\) Theory (1968), Benjamin: Benjamin New York, MR40 #2736 · Zbl 0174.30302
[2] Bautista Ramos, R., Note on isomorphisms of group rings, An. Inst. Mat. Univ. Nac. Autónoma México, 12, 13-18 (1972), MR47 #6822
[3] Bautista Ramos, R., Units of finite algebras, An. Inst. Mat. Univ. Nac. Autónoma México, 16, 1-78 (1976), MR58 #1101 · Zbl 0402.16029
[4] Berman, S. D., Group algebras of countable abelian \(p\)-groups (in Russian), Soviet Math. Dokl., 8, 871-873 (1967), MR35 #5504 · Zbl 0178.02702
[5] Berman, S. D., Group algebras of countable abelian \(p\)-groups (in Russian), Publ. Math. Debrecen, 14, 365-405 (1967), MR37 #1480 · Zbl 0281.20006
[6] Cable, C. A., The decomposition of certain group rings, (Ph.D. Thesis (1969), Pennsylvania State Univ)
[7] Coleman, D. B., On the modular group ring of a \(p\)-group, Proc. Amer. Math. Soc., 15, 511-514 (1964), MR29 #2306 · Zbl 0132.27501
[8] Deskins, W. E., Finite Abelian groups with isomorphic group algebras, Duke Math. J., 23, 35-40 (1956), MR17-1052 · Zbl 0075.23905
[9] Dieckmann, E. M., Isomorphism of group algebras of \(p\)-groups, (Ph.D. Thesis (1967), Washington Univ)
[10] Dubois, P. F.; Sehgal, S. K., Another proof of the invariance of Ulm’s functions in commutative modular g roup rings, Math. J. Okayama Univ., 15, 137-139 (1971/72), MR47 #6892
[11] Eggert, N. H., Quasi regular groups of finite commutative nilpotent algebras, Pacific J. Math., 36, 631-634 (1971), MR44 #262 · Zbl 0197.03201
[12] Fischer, I.; Eldridge, K., Artinian rings with a cyclic quasi-regular group, Duke Math. J., 36, 43-47 (1969), MR38 #5829 · Zbl 0182.05603
[13] Gulliksen, T.; Ribenboim, P.; Viswanathan, T. M., An elementary note on group-rings, J. Reine Angew. Math., 242, 148-162 (1970), MR43 #372 · Zbl 0219.16008
[14] Higman, G., The units of group-rings, Proc. London Math. Soc., 46, 2, 231-248 (1940), MR2-5 · JFM 66.0104.04
[15] Higman, G., Units in group rings, (D. Phil. Thesis (1940), Oxford University) · JFM 66.0104.04
[16] Holvoet, R., Sur les \(Z_2\)-algèbres du groupe diédral d’ordre 8 et du groupe quaternionique, C.R. Acad. Sci. Paris (Sér A-B), 262, A209-A210 (1966), MR32 #5726 · Zbl 0135.05601
[17] Ivory, L. R., A note on normal complements in mod \(p\) envelopes, Proc. Amer. Math. Soc., 79, 9-12 (1980), MR82e:20004 · Zbl 0401.20018
[18] Ivory, L. R., Normal complements in mod \(p\) envelopes, (Ph.D. Thesis (1981), Univ. of Alabama) · Zbl 0401.20018
[19] Janusz, G. J., Faithful representations of \(p\)-groups at characteristic p,l, J. Algebra, 15, 335-351 (1970), MR42 #391 · Zbl 0197.02203
[20] Jennings, S. A., The structure of the group ring of a \(p\)-group over a modular field, Trans. Amer Math. Soc., 50, 175-185 (1941), MR3-34 · Zbl 0025.24401
[21] Johnson, D. L., The modular group-ring of a finite \(p\)-group, Proc. Amer. Math. Soc., 68, 19-22 (1978), MR56 #15744 · Zbl 0264.20020
[22] Kervaire, M. A.; Murthy, M. P., On the projective class group of cyclic groups of prime power order, Comment. Math. Helv., 52, 415-452 (1977), MR57 #16252 · Zbl 0355.12009
[23] Kunz, E., Gruppenringe und Differentiale, Math. Ann., 163, 346-350 (1966), MR33 #200 · Zbl 0141.03704
[24] Laue, R.; Neubüser, J.; Schoenwaelder, U., Algorithms for finite soluble groups and the SOGOS system, (Computational Group Theory (1984), Academic Press: Academic Press London), 105-135 · Zbl 0547.20012
[25] May, W., Commutative group algebras, Trans. Amer. Math. Soc., 136, 139-149 (1969), MR38 #2224 · Zbl 0182.04401
[26] Moran, L. E., The modular group ring of a \(p\)-group, (M. Phil. Thesis (1972), Univ. of Nottingham)
[27] Oliver, R., \( SK_1\) for finite groups rings. I, Invent. Math., 57, 183-204 (1980), MR81f:18031 · Zbl 0428.18011
[28] Passi, I. B.S., Group Rings and Their Augmentation Ideals, (Lecture Notes in Math., 715 (1979), Springer: Springer Berlin), MR80k:20009 · Zbl 1028.20006
[29] Passman, D. S., The group algebras of groups of order \(p\) over a modular field, Michigan Math. J., 12, 405-415 (1965), MR32 #2492 · Zbl 0134.26304
[30] Passman, D. S., The Algebraic Structure of Group Rings (1972), Wiley-Interscience: Wiley-Interscience New York, MR81d:16001 · Zbl 0236.16007
[31] Quillen, D. G., On the associated graded ring of a group ring, J. Algebra, 10, 411-418 (1968), MR38 #245 · Zbl 0192.35803
[32] Raggi Cárdenas, F. F., Units in group rings. I, An. Inst. Mat. Univ. Nac. Autonoma México, 7, 27-35 (1967), MR38 #2225
[33] Roggenkamp, K. W., The isomorphism problem and units in group rings of finite groups, (Groups - St. Andrews 1981 (1982), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 313-327, Zbl.492.66017
[34] Röhl, F., On induced isomorphisms of group rings, (Groups - Korea (1983), Springer: Springer Berlin), to appear · Zbl 0553.16008
[35] Sandling, R., Group rings of circle and unit groups, Math Z., 140, 195-202 (1974), MR52 #3217 · Zbl 0281.20004
[36] Sandling, R., Graham Higman’s thesis “Units in group rings”, (Integral Representation and Applications, 882 (1981), Springer: Springer Berlin), 93-116, MR83g:20009 · Zbl 0468.16013
[37] Sehgal, S. K., On the isomorphism of group algebras, Math. Z., 95, 71-75 (1967), MR34 #5950 · Zbl 0166.02301
[38] Sehgal, S. K., Topics in Group Rings (1978), Marcel Dekker: Marcel Dekker New York, MR80j:16001 · Zbl 0411.16004
[39] Wall, C. T.C., Norms of units in group rings, Proc. London Math. Soc., 29, 3, 593-632 (1974), MR51 #12921 · Zbl 0302.16013
[40] Ward, H. N., Some results on the group algebra of a group over a prime field, (Seminar on Finite Groups and Related Topics (1961), Harvard Univ), 13-19, Mimeo. Notes
[41] Watters, J. F., The adjoint group of a radical ring, (Ph.D. Thesis (1965), Univ. of London) · Zbl 0162.04801
[42] Yu, C. Y., Group rings over modular fields, (Ph.D. Thesis (1970), The Pennsylvania State Univ)
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