To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure firstname.lastname@example.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
A Lie subalgebra L of [gfr ][lfr ][ ](V) is said to be
finitary if it consists of elements of finite
rank. We study the situation when L acts irreducibly on the infinite-dimensional
vector space V and show: if Char [ ] > 7, then L has a unique minimal ideal I.
Moreover I is simple and L/I is solvable.
A group G is said to be a minimal non-FC group, if G contains an infinite conjugacy class, while every proper subgroup of G merely has finite conjugacy classes. The structure of imperfect minimal non-FC groups is quite well-understood. These groups are in particular locally finite. At the other end of the spectrum, a perfect locally finite minimal non-FC group must be a p-group. And it has been an open question for quite a while now, whether such groups exist or not.
A class [Xscr ] of groups is said to be countably recognizable, if every
all of whose
countable subgroups are contained in countable [Xscr ]-subgroups is itself an
Many examples of such classes are discussed in section 8·3 of
. In the present work
we are concerned with the question of how far countable recognizability
obtained for classes of finitary linear groups. Recall that a group is
be finitary [ ]-linear if it is isomorphic to a subgroup of
FGL[ ](V), the group of all invertible
[ ]-linear transformations α of the [ ]-vector space
V with the property that the image
of the endomorphism α−idV has finite
[ ]-dimension. This generalizes the notion of
linearity. A survey about features of finitary linear groups is given
Let be a class of finite groups. Then a c-group shall be a topological group which has a fundamental system of open neighbourhoods of the identity consisting of normal subgroups with -factor groups and trivial intersection. In this note we study groups which are existentially closed (e.c.) with respect to the class Lc of all direct limits of c-groups (where satisfies certain closure properties). We show that the so-called locally closed normal subgroups of an e.c. Lc-group are totally ordered via inclusion. Moreover it turns out that every ∀2-sentence, which is true for countable e.c. L-groups, also holds for e.c. Lc-groups. This allows it to transfer many known properties from e.c. L-groups to e.c. Lc-groups.
In this paper, will always denote a local class of locally finite groups, which is closed with respect to subgroups, homomorphic images, extensions, and with respect to cartesian powers of finite -groups. Examples for x are the classes L ℐπ of all locally finite π-groups and L(ℐπ ∩ ) of all locally soluble π-groups (where π is a fixed set of primes). In , a wreath product construction was used in the study of existentially closed -groups (=e.c. -groups); the restrictive type of construction available in  permitted results for only countable groups. This drawback was then removed partially in  with the help of permutational products. Nevertheless, the techniques essentially only permitted amalgamation of -groups with locally nilpotent π-groups. Thus, satisfactory results could be obtained for Lp-groups (resp. locally nilpotent π-groups) , while the theory remained incomplete in all other cases.
We study the embeddings of a finite p-group U into Sylow p-subgroups of Sym (U) induced by the right regular representation p: U→ Sym(U). It turns out that there is a one-to-one correspondence between the chief series in U and the Sylow p-subgroups of Sym (U) containing Up. Here, the Sylow p-subgroup Pσ of Sym (U) correspoding to the chief series σ in U is characterized by the property that the intersections of Up with the terms of any chief series in Pσ form σp. Moreover, we see that p: U→ Pσ are precisely the kinds of embeddings used in a previous paper to construct the non-trivial countable algebraically closed locally finite p-groups as direct limits of finite p-groups.
A theorem of G. Higman about the embeddability of amalgams within the class of all finite ρ-groups is generalized to classes of soluble groups. We also give best possible bounds for the solubility lengths of the constructed completions. And, as an application, the super-soluble amalgamation bases in the class of all finite soluble π-groups are determined.