The thesis is devoted to the study of faithful finite automaton representations
of finitely generated groups. Residual finiteness is a natural necessary condition
for groups that allow such representations.
Faithful finite automaton representations of finitely generated residually finite
groups, which are close to free groups in their combinatorial structure, are consi-
dered. More precisely, it is considered groups that decompose into free products
or amalgamated free products of their subgroups, and HNN extensions of groups.
Chapter 2 is devoted to the construction of faithful representations of amal-
gamated free products of finite cyclic groups with cyclic subgroup amalgamated
by finite automata over some finite alphabet. Namely, using a generalization of
the ping-pong lemma, it is shown that the specified amalgamated free product is
generated by automaton permutations defined by initial automata with 4 states.
The case of trivial amalgamation is considered separately, i.e. a faithful repre-
sentation of the free product of finite cyclic groups is constructed. In this case,
the automata in which the necessary automaton permutations are defined have 2
internal states.
In Chapter 3 a faithful representation of amalgamated free products of cyclic
groups with cyclic subgroup amalgamated by finite automaton permutations over
the minimal possible alphabet is constructed. The auxiliary construction of auto-
maton permutations with arbitrary finite order is given and the number of states
in the automaton that define the specified automaton permutation is estimated.
For the explicit construction of automata, in the states of which the automaton
permutations required for the proof will be determined, the operations of linking
and serial linking of two initial automata are used, as well as the construction of an
l-complete automaton. In the case of faithful representations of amalgamated free
products of finite number of cyclic p-groups by initial automata, it is additionally
shown that such an amalgamated free product is isomorphically embedded in the
group p-F GA(X) over an alphabet X of cardinality p, i.e. the intersection of the
p-Sylow subgroup Sylp(GA(X)) of the group of all automaton permutations over
X and the group F GA(X) of all finite automaton permutations over X. The proof
of the last statement is based on the general construction proposed above, and on
additional analysis of the automaton constructed in it.
Chapter 4 considers the question of finding HNN extensions of free abeli-
an groups of rank n, n ≥ 1, which allow isomorphic embedding into the group
p-F GA(X). It is introduced HNN extensions defined by invertible integer matri-
ces M of size n × n with some additional conditions. It is shown that these
HNN extensions of free abelian groups allow isomorphic embedding in the group
p-F GA(X). As a result, it is obtained that the corresponding HNN extensions
are residually p-finite. It is presented an example of a matrix of size 2 × 2, such
that it does not satisfy the previously defined conditions, but corresponding HNN
extension allow an embedding into 2-F GA(X).
The main results that determine the scientific novelty of the thesis are as
follows.
1. Faithful finite automaton representations are constructed for amalgamated
free products of finite number of finite cyclic groups with amalgamated cyclic
subgroup. The automaton permutations that generate corresponding products
are defined by initial automata, each of which has 4 internal states.
2. Faithful finite automaton representations of free products of finite cyclic
groups are constructed. The automaton permutations that generate corres-
ponding products are defined by initial automata, each of which has 2 internal
states.
3. For amalgamated free products of finite number of finite cyclic groups with
amalgamated cyclic subgroup, faithful finite automaton representation are
constructed over a minimal possible alphabet.
4. For arbitrary prime number p it is proved that amalgamated free products
of finite number of cyclic p-groups with amalgamated cyclic subgroup are
represented by finite automata of special form. It is shown that corresponding
automaton permutations belong to the p-Sylow subgroup of the group of all
automaton permutations over the alphabet with p elements.
5. A class of HNN extensions of free abelian groups of finite rank is introduced.
All groups from this class are represented by finite automaton permutations. It
is proved that corresponding automaton permutations belong to the p-Sylow
subgroup of the group of all automaton permutations over the alphabet with
p elements.
The thesis is a theoretical research. The obtained results can be used in
algebra, automata theory, discrete mathematics, algorithm theory and other fields
of knowledge, the methods of which are based on automaton representations of
algebraic structures.
