LATTICE UNIVERSALITY OF LOCALLY FINITE p -GROUPS Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 9, No. 1, 2023, pp. 127-134

DOI: 10.15826/umj.2023.1.011

LATTICE UNIVERSALITY OF LOCALLY FINITE p-GROUPS

Vladimir B. Repnitskil

Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russian Federation

[email protected]

Abstract: For an arbitrary prime p, we prove that every algebraic lattice is isomorphic to a complete sublattice in the subgroup lattice of a suitable locally finite p-group. In particular, every lattice is embeddable in the subgroup lattice of a locally finite p-group.

Keywords: Subgroup lattice, Algebraic lattice, Complete sublattice, Lattice-universal class of algebras, Locally finite p-group, Group valuation.

1. Introduction and formulation of results

A lattice is called algebraic if it is complete and each element is a join of compact elements. Important examples of algebraic lattices include subalgebra lattices of universal algebras, particularly subgroup lattices of groups.

If a lattice L is complete and a subset M of L has the property that VS, /\S € M for every nonempty subset S C M, then M is called a complete sublattice of L. It is well known that a complete sublattice of an algebraic lattice is itself algebraic. This implies that complete sublattices of subgroup lattices are algebraic as well. Conversely, we proved in [7] that every algebraic lattice can be represented as a complete sublattice of the subgroup lattice of a suitable locally finite 2-group. It should be noted that every lattice can be embedded in some algebraic lattice, namely, in the lattice of its ideals. It follows that every lattice is embeddable in the subgroup lattice of a locally finite 2-group. For a given class K of algebras, we say that K is lattice-universal if every lattice is embeddable in the subalgebra lattice of some algebra from K. In this sense, the class of all locally finite 2-groups is lattice-universal. It must be said here that the lattice universality of the class of all groups was first proved by Whitman in [12]. Other examples of lattice-universal classes of algebras can be found in [3-5, 10].

The main theorem of the present paper generalizes the key result of the author's paper [7] from the case p = 2 to the case of an arbitrary prime number p.

Theorem 1. For an arbitrary prime p, let K be an abstract class of groups satisfying the following conditions:

(1) K contains a group of order p;

(2) K is closed under restricted direct products, semidirect products and direct limits over totally ordered sets.

Then every algebraic lattice is isomorphic to a complete sublattice in the subgroup lattice of some group in K.

Since the class of all locally finite p-groups obviously satisfies conditions (1)-(2) of the theorem, from here we get the following corollary.

Corollary 1. For an arbitrary prime p, every algebraic lattice is isomorphic to a complete sublattice in the subgroup lattice of a .suitable locally finite p-group.

As a consequence, we get the following statement.

Corollary 2. For every prime p, the class of all locally finite p-groups is lattice-universal.

The proof of Theorem 1 is given in the next section. Technically, it is based on the concept of so-called group valuations used by us in [7] when proving a specific case of this theorem for p = 2. Here we essentially apply the ideas and constructions of the mentioned paper. Below we give some additional concepts and notation. Sub G is the subgroup lattice of a group G. (X} is the subgroup generated by a subset X of a given group.

The commutator [u, v] of elements u and v of a group means u-1v-1uv and uv means v-1uv.

n GA is the direct product of a set (GA | A € A} of groups.

AgA

n G\ is the restricted direct product of a set {Ga | A € A} of groups; this group is a subgroup AeA

of the corresponding direct product, and it can be regarded as the group of all functions from

fun(A, (J Ga) with the property f (A) € Ga and with finite supports. AeA

For given groups G and T, let us consider the direct product H Gh of isomorphic copies Gh

heT

of the group G; here G1 = G. This group is regarded as the group fun(T, G) of all functions from T to G and is denoted by GT. The group T naturally acts on the group

GT

in the following way:

f *(h) = f (th) for all f € GT and t € T. With respect to this action, one can consider the semidirect product T X GT, which is denoted by GIT and is called the wreath product of the group G by the group T. Here, for t1,t2 € T and f1, f2 € GT, we have

(tl,fl) ■ (t2,f2) = (t1t2, f12f2).

For a prime p, the group

(u, v | up = vp = [u, v]p = 1, [u, [u, v]] = [v, [u, v]] = 1)

is of the order p3 and is isomorphic to the multiplicative unitriangular group of matrices of order 3 over the field of order p. So it will be denoted by UT3(p). Here the generating elements u and v correspond to the matrices

1 1 0 \ / 1 0 0 \ 0 10 1 and 0 11 , 0 0 1 0 0 1

respectively. For p = 2, this group is isomorphic to the 8-element dihedral group

D4 = (u,v | u2 = v2 = (uv)4 = 1).

Obviously, we have

UT3(p) = (u} X (v, [u, v]},

where the group (u} acts by conjugation on the normal subgroup (v, [u, v]} in UT3(p) and

(v, [u, v]} = (v} x ([u, v]}.

J(P) is the ideal lattice of a given V-semilattice P with zero. Recall that every algebraic lattice is isomorphic to the ideal lattice J(P) of its V-semilattice P of compact elements.

The other definitions and notations used but undefined in the paper can be found in the books [1, 2, 9].

2. Concept of group valuation and the proof of the theorem

Let us give a key notion of the present paper. It was introduced by the author in [6] (see also [7, 8, 11]).

Definition 1. Let P be a V-semilattice with zero. For a group G, we call a mapping 5: G — P a group valuation if the following conditions hold:

(1) 5(1) =0;

(2) 5(g-1) = 5(g) for every g € G;

(3) 5(g1 g2) < 5(g1) V 5(g2) for every g1,g2 € G.

For an arbitrary ideal I € J(P) and a € P, let us put Os(I) = 5-1 (I) and Os(a) = Os(al), where al denotes the principal ideal generated by the element a. Obviously, Os (I) is a subgroup of G and Os (I) = V aei Os (a).

The following simple proposition explains the role of the notion of group valuation in our examination (see its proof, for example, in [7]).

Proposition 1. Let a valuation 5: G — P satisfy the following conditions:

(1) for every a,b € P and g € G, 5(g) < a V b implies g € (Os(a), Os(6));

(2) 5 is a surjective mapping of G onto P.

Then the mapping Os: J(P) — Sub G is a complete embedding of the ideal lattice J(P) in the subgroup lattice Sub G, and so the lattice J(P) is isomorphic to a complete sublattice of the corresponding subgroup lattice.

For groups G, G' and their valuations 5: G — P, 5': G' — P, we say that the pair (G',5') is an extension of the pair (G, 5) if G is a subgroup of G' and 5'|G = 5.

The following statement is key in proving our theorem.

Proposition 2. Let G be a group and 5: G — P a group valuation. Then, for arbitrary elements a,b € P, there exists an extension (G', 5') of the pair (G, 5) such that, if 5(g) < a V b for an element g € G, then in the group G' the membership g € (Os'(a),Os'(b)) holds; moreover, for an arbitrary prime p, one may take the group G I UT3(p) as G'.

This statement was proved in [7] for p = 2, and in this case the role of the group UT3(p) was played by the dihedral group isomorphic to UT3(2).

We will prove Proposition 2 at the end of the section. But now we will explain how the theorem is derived from it. This derivation practically iterates a similar derivation for p = 2 in our paper [7], but we will give it for the reader's convenience.

Let L be an arbitrary algebraic lattice and P its V-semilattice of compact elements. Thus, we have L = J(P). Let p be an arbitrary prime number and K an abstract class of groups satisfying conditions (1)-(2) of the Theorem 1. Now consider the set {(wa) | a € P} of groups of order p generated by the elements wa indexed by elements from P. Let

g* = n («0

a€P

be the corresponding restricted direct product of these groups. Then G* € K and each non-identity element from G* can be uniquely represented (up to the permutation of the factors) as a term of

the form w^lw^, ■ ■ ■ w^n (here n > 1, wa = waj for i = j and 1 < e» < p — 1). Now we define a mapping S* : G* —> P by the following rule: S*(1) = 0 and, if g = w^lw^ ••• w^n, then S*(g) = a1 V a2 V ■ ■ ■ V an. It is easy to see that the mapping S* is a group valuation. Now let

{(a7,b7) € P x P | 0 < Y < X} be the well-ordered set of all couples from P2. We define by induction on y a set

(G7 | 0 < y < X}

of groups from K and a set

(S7 | 0 < y < X}

of group valuations SY: GY ^ P in the following way.

Put Go = G* and So = S*. If the ordinal y > 0 is not limit, then the pair (GY, SY) is an extension of the pair (GY-1,SY-1) with the property g € (O<s7 (a7),O<s7 (bY)} in GY for every g € G* satisfying the condition SY-1 (g) = S* (g) < a7 V bY. Such an extension exists by Proposition 2, and in this case GY = GY-11 UT3(p). If the ordinal y is limit, then put

Gy = U(Gc I Z < Y) and SY = [J(SC | Z < y).

Further, put

G(1) ^(G7 | Y < x) and S(1) = U(S7 | y < x).

By construction, the mapping S(1): G(1) ^ P is a valuation and the pair (G(1), S(1)) is an extension of the pair (G*,S*) such that, for any elements g € G* and a, b € P with S*(g) < a V b, the membership g € (O^i) (a), O5(i) (b)} holds in G(1). In addition, we have UT3(p) € K, since K contains all p-element groups and, by condition (2) of the theorem, is closed under direct and semidirect products of any two of its groups and UT3(p) as mentioned above is a semidirect product of the p-element group (u} with the direct product of the p-element groups (v} and ([u, v]}. Therefore, G(1) € K because, by the same condition, the class K is closed also under direct limits over totally ordered sets.

Next we construct the following increasing under inclusion chains of groups G(n) € K and valuations S(n): G(n) ^ P (n € N):

G* = G(0) c G(1) c G(2) c ..., S* = S(0) c S(1) c S(2) c ...

such that, for every n > 1, S(n) |G(n-1) = S(n-1) and, for every g € G(n-1) and a, b € P, if S(n-1)(g) < a Vb, then g € (O^n) (a),O5(n) (b)} in G(n)

; here the pair (G(n), S(n)) is constructed from the pair (G(n-1), S(n-1)) as it was done in the first step corresponding to n = 1. Put

G = [_J(G(n) | n € N) and S = [J(S(n) | n € N).

Then, by construction, we have G € K and the mapping S: G ^ P is a valuation with the property: for all g € G and a, b € P, the inequality S(g) < a V b implies g € (O<s(a),O<5(b)}, i.e., S satisfies condition (1) of Proposition 1. In addition, S* is surjective and S|G* = S*, whence we deduce that the valuation S is surjective as well, i.e., it satisfies condition (2) of Proposition 1. Therefore, by this proposition, there exists a complete embedding of the ideal lattice J(P) in the subgroup lattice Sub G of the constructed group G € K. This means that the algebraic lattice L, for which P is the V-semilattice of compact elements, is isomorphic to a complete sublattice of Sub G. The derivation is over. □

The following two constructions of the semidirect product and direct product of group valuations were introduced by the author in [7].

If a group T acts on a group H, and if 5: T — P and 5: H — P are group valuations such that

5(ht) < 5(t) V 5(h)

for any h € H and t € T, then the mapping 5 A 5: T A H — P defined by

(5 A 5) (t, h) = 5(t) V 5(h)

is evidently a group valuation. It extends both 5: H — P and 5: T — P under the canonical isomorphic embeddings h — (1, h) and t — (t, 1) of H and T respectively into T A H. Thus the pair (T A H, 5 A 5) is an extension both of the pair (H,5) and of the pair (T,5).

Definition 2. The valuation 5 A 5 is called the semidirect product of 5 and 5.

Definition 3. If 5i: Gi — P, i = 1,...,n, are group valuations, then the mapping 51 x ■ ■ ■ x 5n: G1 x ■ ■ ■ x Gn — P defined by

(51 x ■ ■ ■ x 5n)(g1,...,gn) = 51(g1) V ■ ■ ■ V5n(gn)

is evidently a group valuation extending each 5i under the canonical embedding of Gi into G1 x ■ ■ ■ x Gn. It is called the direct product of the valuations 5i's.

Let G be a group, 5: G — P be a valuation, and let a,b € G. Consider the group

UT3(p) = (u,v I up = vp = [u,v]p = 1, [u, [u,v]] = [v, [u,v]] = 1).

Each of its elements can be uniquely written in the form uav3[u, v]Y, where a, € {0,1,... ,p-1}. It is easy to check that the product of two such terms is

uai v31 [u,v]Y1 ■ ua2 v32 [u,v]Y2 = uai+a2 v3l+32 [u,v]Y1 +Y2-31«2 ,

where exponents are added and multiplied modulo p. Now we define a mapping 5: UT3(p) — P by the rule:

5(1) = 0, 5(ua) = a if 1 < a < p - 1, 5(v3) = b if 1 < p < p - 1, 5(t) = a V b for all other t € UT3(p).

It is easy to see that the mapping 5 is a group valuation.

Next we construct an extension 5': G I UT3(p) — P of a group valuation 5 desired in Proposition 2 from the group valuation 5: UT3(p) — P and from an additional valuation 5: GUT3(p) — P. The valuation 5 will be the direct product (see Definition 3) of group valuations 5t: G — P defined below for every t € UT3(p). Their constructions generalize the corresponding constructions for p = 2 given in [7].

We set 5t(1) = 0 for any t € UT3(p). For a non-identity element g € G, we set

(1) 51(g) = 5(g),

(2) 5ua (g) = a V 5(g) if 1 < a < p - 1,

(3) 5vt3 (g) = b V 5(g) if 1 < p < p - 1,

(4) S (g) = <0 if S(g) < a V b, if 1 < y < p 1

[u'v]7 la V b V S(g) otherwise _ _ '

. la if S(g) < a V b,

(5) S„«[„,v]y (g) = < . if 1 < a < p — 1 and 1 < Y < p — 1,

la V b V S (g) otherwise

, x c- f b if S(g) < a V b,

(6) Sv^v^g) = . . - if 1 < P < p — 1 and 1 < Y < p — 1,

la V b V S(g) otherwise

(7) Smav/3 [u>v]7 (g) = a V b V S(g) if 1 < a < p — 1, 1 < p < p — 1 and 0 < y < p — 1.

Checking that St is a group valuation for every t € UT3(p) is very simple and completely identical to checking a similar statement in [7].

Lemma 1. For any w,t € UT3(p) and g € G, the inequality Stw(g) < ¿(t) V Sw(g) holds.

Proof. From the definitions of the group valuations St, it directly follows that the equality a V b V Sw(g) = a V b V S(g) holds for every w € UT3(p) and g € G. Therefore, if ¿(t) = a V b, then we have

Stw (g) < a V b V S(g) = a V b V Sw (g) = ¿(t) V Sw (g). The case ¿(t) = a V b is true only if

t € {1, ua, v^ | 1 < a < p — 1, 1 < p < p — 1}.

In the case t = 1 the inequality Stw(g) < ¿(t) V Sw(g) is evident.

Let t = ua, where 1 < a < p — 1. Then ¿(t) = a. Here, if b < Sw(g), then again

Stw (g) < a V b V S(g) = a V b V Sw (g) = ¿(t) V Sw (g).

If b < Sw (g), then the following four cases are possible:

(1) w = 1,

(2) w = ua', where 1 < a' < p — 1 and Sw (g) = a V S(g),

(3) w = [u, v]Y, where 1 < y < p — 1, S(g) < a V b and Sw(g) = 0,

(4) w = ua' [u, v]Y, where 1 < a' < p — 1, 1 < y < p — 1, S(g) < a V b and Sw (g) = a.

In case (1), we have Siw(g) = SM«(g) = a V S(g) = a V S1(g) = S(t)V Sw(g). In case (2), we have Siw (g) = (g) < a V S(g) = a V Sw (g) = ¿(t) V Sw (g). In case (3), we have Siw (g) = Sua[u;v]7 (g) = a < a V Sw (g) = ¿(t) V Sw (g). In case (4), we have Siw (g) = [u>v]7 (g) < a = ¿(t) V Sw (g).

The case t = v^, where 1 < p < p — 1, is symmetric to the previous one and can be checked by a parallel argument. □

Next, as noted above, we set S: GUT3(p) ^ P as the direct product of the valuations St, t € UT3(p). This is a group valuation extending S1 = S and

«(/)= V St(f (t))

teUT3(p)

for any f: UT3(p) — G.

We then define 5': G I UT3(p) — P as the semidirect product of 5: UT3(p) — P and 5: GUT3(p) — P (see Definition 2). This is an extension of 5, hence also of 5. To prove that the semidirect product 5' = 5 A 5 is indeed a group valuation, we have to check that

5 (ft) < 5(t) V 5 (f)

for any f € GUT3(p) and t € UT3(p). But this is easily accomplished using Lemma 1:

5 (ft) = V 5w (ft(w)) = V 5w (f (tw)) = V 5t-1w (f (tt-1w))

weUT 3(p) weUT3(p) t-1weUT3(p)

V 5t-1w(f (w)) < V (5(t-1) V 5w(f (w)))

^ ^ 1 1 i n 1 1 v \/ i/iit 1)

t-1weUT3(p) t-1weUT3(p)

= 5(t-1) V V 5w(f (w)) = 5(t) V V 5w(f (w))

\t-1weUT3(p) / \weUT3(p)

= 5(t) V 5(f).

Thus, we proved that the mapping 5': G I UT3(p) — P is a group valuation and the pair (G I UT3(p),5') is an extension of the pair (G,5). Now it remains to note that this extension has the required property, i.e., if 5(g) < a V b for an element g € G, then, in the group G I UT3(p), the membership g € (Os'(a),Os'(b)) holds. Indeed, by construction, 5'(u) = 5(u) = a and 5'(v) = 5(v) = b. In addition, the element g € G as an element of the group G I UT3(p) means a function

f (w) € GUT3 (p)

such that

g, if w = 1, f (w) = t 1, if w = 1.

Therefore, for an arbitrary t € UT3(p), we have gt = f t(w) = f (tw) and

5' (gt ) = 5(ft)= V 5w (f t(w))= V 5w (f (tw))

weUT 3(p) weUT3 (p)

= 5t-1 (f (1)) V V 5w(f (tw)) = 5t-1 (g) V V 5w (1)1 = 5t-1 (g).

\w=t-1 / \w=t-1 J

From here we obtain that 5'(g) = 5[u v](g) = 0 for every element g € G such that 5(g) < a V b. This implies that

g = [v,u] ■ g[v,u] ■ [u,v] € (Os'(a),Os'(b),Os'(0)) = (Os'(a),Os'(b)).

The proof of Proposition 2, and with it the Theorem 1, is complete. □

Acknowledgments

The author wishes to thank Mikhail Volkov and anonymous referees for their comments and kind help in preparing the final version of this paper.

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