By Alfonso Di Bartolo, Giovanni Falcone, Peter Plaumann, Karl Strambach

Algebraic teams are handled during this quantity from a gaggle theoretical viewpoint and the got effects are in comparison with the analogous concerns within the concept of Lie teams. the most physique of the textual content is dedicated to a category of algebraic teams and Lie teams having in simple terms few subgroups or few issue teams of other sort. specifically, the variety of the character of algebraic teams over fields of optimistic attribute and over fields of attribute 0 is emphasised. this can be published by way of the plethora of three-d unipotent algebraic teams over an ideal box of confident attribute, in addition to, by means of many concrete examples which hide a space systematically. within the ultimate part, algebraic teams and Lie teams having many closed common subgroups are determined.

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**Example text**

As n + 1 ≤ pt , we have t t M p − I = (M − I)p = 0. On the other hand, for x0 = 0 the rank of the matrix (M − I) is n. Thus the dimension of the eigenspace relative to the (unique) eigenvalue 1 is one and this forces the minimal polynomial of M to be equal to the characteristic (t−1) = 0. polynomial, hence (M − I)p We note that for p > n, the group Jn (α) provides an example of an n-dimensional group of nilpotency class n where every element has exponent p. In the class of ﬁnite p-groups to these groups there correspond ﬁnite groups of exponent p and of nilpotency class p − 1.

Xn−2 . In fact the matrices above form a group, the centre of which is the one-dimensional subgroup consisting of those matrices, where only a1,n is diﬀerent from zero. 10 Proposition. If r = s and n > 1, then there is no surjective homomorphism from Jm (Fr ) to Jn (Fs ). Proof. In order to prove the assertion, we ﬁrst assume that m = n = 2. Let γ : J2 (Fr ) −→ J2 (Fs ) be a homomorphism and put γ(x0 , x1 ) = γ((x0 , 0)(0, x1 )) = γ(x0 , 0)γ(0, x1 ) = (γ1 (x0 ), γ2 (x0 ))(0, γ3 (x1 )) = (γ1 (x0 ), γ2 (x0 ) + γ3 (x1 )).

593, that in the semigroup of isogenies of a Witt group the Ore condition holds. For a vector group this follows from [54], § 10, p. 313. Hence can ﬁnd two isogenies hi : Ai −→ A such that h1 f1 = h2 f2 and we obtain h2 [φ2 ]g2 = h1 [φ1 ]g1 . We have already mentioned that the groups SL2 and P SL2 show that the existence of an isogeny is not a symmetric relation. However, if there exists an isogeny from a connected commutative unipotent group G1 onto G2 then an isogeny from G2 onto G1 exists as well (see [89], Proposition 10, p.