Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

SZAKÁCS, A., Unitary subgroup of the Sylow 2-subgroup of the group of normalized units in an infinite commutative group ring

Unitary subgroup of the Sylow 2-subgroup of the group of. 93 because the elements of the such constructed set M can be considered as the representatives of the cosets of the group W

2 (KG)[2] by the subgroup W

2" (KG)[2]. Note that the elements of the set M we shall choose in the form yy* (y G V

2" (KG))

. Let A) holds, i.e. |K

n| > \G

n\. It is easy to prove that in this case the Sylow 2-subgroup S

n of the group G

n has such elfement g of order 2 and there exists an a £ G

n that one of the following conditions holds: Ai) G

n / (g),a $ (g) and a

2 (g), A

2) Gn / (g),d $ (g) and a

2 G (g), A

3) G

n = (g) and in cases Ax

) and A

2) at least one of the elements a or g do not belong to the subgroup G

n+\

. Indeed, if g G G

n+i then, by condition G

n / G

n+1, the set G

n \ G

n+1 has a proper element a. Let Ai) holds. Let a be a nonzero element of the ring K

n and y

a = 1 -f aa( 1 4- g). We shall prove that the set M = {x

a - y

ay

a* = 1 + a(a + a"

1)(

1 +d)' 0 / a G K

n} has the above declared property. Really, since a

2 (g), it follows that the elements a and a~

l belong to the different cosets of the group G

n by the subgroup (g). Hence x

a ^ 1. It is easy to see that x

a* = x

a = x

a~~

l

. Therefore x

a is a unitary element of second order of the group V(K

nG

n). If x

a G V

2 then, from the condition a

2 (fc (g

), it follows that the elements a and ag belong to the group G

n+1

, but this contradicts to the choice of elements a and g. Therefore z

a G W

2" [2] \ W

2n+

1[2}. Suppose that the coset x

aV

2

n [2] coincides with x

v V

2 [2] for a different a and u from K

n. Then x

a = x

uz for a suitable z G V

2" . Since x„* = Xi,1, it follows that z - x

ax„* = 1 + (a + u)(a + a1)(l + g) = x

a+l

/ 'and x

a+l

/ belongs to the subgroup y

2"

+

1 what contradicts it which was proved in above. Obviously \M\ = \K

n\. Therefore the constructed set M has the above declared property. Let A

2) holds. It is easy to see that the elements of the set M = {x

a = 1 + aa( 1 + g)

: 0 ^ a G K) belong to the different cosets of the group V(KG)[

2] by the subgroup V

2(KG)[

2]. Indeed, if x

a G V

2 then a G Gi and ag G G\. But this contradicts to the choice of the elements a and g and hence x

a £ W[

2] \ VF

2 [2].