On the balanced subgroups of modular group rings
Автор: Danchev Peter V.
Журнал: Владикавказский математический журнал @vmj-ru
Статья в выпуске: 2 т.8, 2006 года.
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The balanced property of certain subgroups of the group of all normalized p-torsion invertible elements in a modular group ring of characteristic p is explored.
Короткий адрес: https://sciup.org/14318180
IDR: 14318180
Текст научной статьи On the balanced subgroups of modular group rings
Let S ( RG) be the normed p-unit group in a group ring RG, formed by an abelian group G and a commutative ring R with identity of prime characteristic p . All unexplained symbols and letters as well as the terminology and definitions from the abelian group theory (including the topological ones) can be found in the classical book monographs [7]. For a background material in that direction, we refer the reader also to [1]–[6].
The major goal motivating the present paper is to find some special nice and isotype subgroups of S ( RG), a problem that arises naturally in the examination of the total projectivity both in modular and semi-simple aspects (cf. [1] and [6]). Thus the property of subgroups being balanced in modular group rings is crucial for the investigation of nice composition series and nice bases in such rings (see, for instance, [9] or [5]).
Moreover, the balanced subgroups play an important role for the quasi-completeness (e. g. [2, 3]) and torsion-completeness (e. g. [4]) in group algebras by using either an algebraical or topological technique in terms of bounded convergent Cauchy sequences.
The query for the balanced property of S ( KH ) in S ( KG) when KG is semisimple, such that G is p -primary and K is either a field having arbitrary characteristic or is a special ring of zero characteristic, is considered and settled in some way by us in [4].
In [9] and [8], May and Hill-Ullery studied the case when R is a field, whereas we here investigate the general situation which cannot be treated by similar reasons.
The main result
We start with a single key assertion needed for future applications. It discovers the balanced property in S ( RG) of subgroups of the type S ( RH ), whenever H 6 G; for certain other balanced subgroups the readers can see [6].
Proposition. Let H be a p -balanced ( that is p -nice and p -isotype ) subgroup of G . Then S ( PH ) is balanced in S ( RG) , provided P is a perfect subring of R with the same unity.
C « p -nice» . Bearing in mind [7], it is enough to calculate that Q [S p a ( RG)S ( PH )] = α<τ
S p ( RG)S ( PH ) for every limit ordinal т . In fact, given an element x in the left hand-side,
mnm hence, by [3], x G (P rigi)S (PH) = (P ri gi )S (PH) = ..., where ri £ Rp, P ri = 1, i=1 i=1
α βn gi G Gp ; ri G Rp , P ri = 1, gi G Gp ; a < в < т and в is arbitrary but a fixed ordinal. i=1
Thus we can write
m rigi = i=1
Xn ri0gi0 Xn fihi i=1 i=1
= E Er i f g i h j , ij
n whenever fi G P with fi = 1 and hi G H .
i =1
Writing ri0 fj gi0 hj in canonical form, we may presume without loss of generality that the i,j following relations hold:
0 0 0 0 00 0
r i J i = 0, r 1 f 2 = • • • = r i J n = 0; r 2 f 2 = 0, r 2 J i = r 2 f 3 = • • • = r 2 f n = 0; • • • ;
r S f s = °, r S f i = • • • = r S f s- i = r S f s +1 = • •• = r S f n = 0
for some s ∈ N , and all other ring products are not zero. Of course, these ring dependencies are indeed correct and well-chosen, because if in addition rf i = 0 we detect that 0 = r ^ f + • • • + f n ) = r i which is a contradiction. Moreover, we note that r i f i = r i (f i + • • • + f n ) = r ( , • • •, r S f s = r S (f i + • • • + f n ) = r S .
Now, let us assume for difficulty that the following additional group ratios hold (if not, the things are easy): g 2 0 h 2 = g 3 0 h 3 = • • • = g S 0 - 1 h S - 1 such that r 2 0 f 2 + r 3 0 f 3 + • • • + r S 0 - 1 f S - 1 = 0, i. e. these elements do not lie in the support.
A crucial fact is that, since the supports of the elements in the group ring are finite while the set { a < в < т : в > ш } is infinite, all given relations are assumed of the above types presented. We mention that all other variants, even when there is no zero divisors, are identical or have a simple interpretation.
The canonical records imply ri = rf gi = g0hi; r2 = rS+iA, g2 = gS+ihi; r3 = rS+2/2, дз = gS+2h2; • • •;
r k = r S +i f 2 , g k = g S +i h 2 r k +i = r S +2 Л, g k +i = g S +2 h i ; • • •; r S = r S f s , g S = g S h s •• •;
r n = r n f n , g n = g n h n ; • • • ; r m- 2 = r n - 2 f n- i , g m- 2 = g n - 2 h n- i ;
r m- i = r n - i f n i g m- i = g n - i h n ; r m = r nfl i g m = g n h i ^
Therefore, we get that, ri G T Rpe = RpT, • • •, rm G RpT, hence ri G RpT, • • •, гП G RpT since β<τ r'i = r(fi = ri, • ••, rS = rS/s = rS, rS0 +1 = rS0 +1f1 + • • • + rS0 +1fn = r2 + • • • ,
..., rn = rn f1 + • • • + rn fn = rm + • • • + rn , where m = n2 — s + 2 — s(n — 1) = n2 — sn + 2. Besides, gi G \ (Gpe H) = GpT H, •••,gm G GpT H^
β<τ
Thus we can write g i = g Ti a i , ..., g m = g Tm a m where g T i ,..., g Tm E G p T and a i ,..., a m E H . Since g i g - i E G p T , whence a i a - i E G p T , we shall presume that a i = a ^ because g Ti a i = g T 022 for some g T i E G p . By the same token we may produce also for the other pairs of indices ( i, j ) such that g i g-1 E G p . Besides, g i g - = h i h - E H , hence g Ti g - E H . The same procedure can be done for the other pairs of distinct indexes with this property as well.
mnn
We observe that 52 rigi = I 52 rigTui I 52 fibi , where for 1 6 i 6 n we have bi = aui i=i i=ii=i or so
bi = auigTvigTWi E H for some appropriate permutations Ui, Vi, Wi of the indexes 1,..., n that gT2b2 = gTзЬз = • • •
= д т ( s- i) b s- i , and eventually r i = r u i .
mm
When m > n it may be possible that 52 rigi = I 52 rigTi ) a for some a E H• i=i Vi=i/
m
Since 52 r i g i E S(RG), there exists a group member from the sum which member belongs i =1
nn to Gp . By a reason of symmetry the same should be valid even for ri0 gi0 and fi hi . So, i=i i=i with no harm of generality, we may suppose that: gi,... ,gi E Gp, ri + ... + ri — 1 belongs to the nil-radical of R; Gp 3 gi+i E gi+2GP E ... E gmGP, ri+i + ri+2 + • • • + rm lies in the nil-radical of R; l E N. Analogously gi,... ,g'k E Gp, ri + ... + rk — 1 belongs to the nil-radical of R; Gp 3 gk+i E gk+ 2Gp E ... E g'nGp, rk+i + rk+2 + ... + rn lies in the nilradical of R and hl,... ,hk E Hp, fi + ... + fk — 1 is in the nilradical of R; Hp 3 hk+i E hk+2 Hp E ... E hnHp, fk+i + fk+2 + ... + fn is in the nilradical of R; n E N.
Because, for any ordinal 5, we know that ( G p H ) p = G p H p , we will presume that g T i E G pp T and a 1 E H p . Moreover, by what we have already proved,
ττ τ τ gl+igl+2 E (G H)P = Gp HP, . . . , gl+igm E Gp Hp, . . . , gl+2gm E Gp Hp etc. gk+1gk+2 E Gp Hp , . . . , gk+1gn E Gp Hp , . . . , gk+2gn E Gp Hp etc.
Similarly for h k+i h - +2 E H p , ..., h k+i h-1 E H p , ..., h k+2 h-1 E H p etc.
Furthermore, b i = a u i g Tv i g ^W i E H p for i = 1,... ,k or b i E b j H p for k + 1 6 i = j 6 n.
Finally, it is apparent that PP r 0 g Tu i E T S ( Rp< a G p " ) = S ( R p G p ) = S p T (RG) and i =i а<т
PP f i b i E S ( PH ). That is why, it is easily checked that x E S p T ( RG ) S ( PH ). Thereby, the i=1
wanted equality is true, as expected.
« p -isotype» . Exploiting [1],
S ( PH ) П S p “ (RG) = S ( PH ) П S ( RP“Gp“ ) = S (P(H П G p “ )) = S ( PH p “ ) = S p “ ( PH ) .
So, the proof is completed in all generality. B
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