In the sixties S.Gähler ([8] and [9]) introduced and studied the basic properties of 2 -metric and 2 -normed spaces. Since then these topics have been intensively studied and deve loped.The references given at the end of this paper are far from being complete, containing only the papers related to the problems treated here.

The aim of the present paper is to study the relations between the extension properties of bounded bilinear functionals and the approximation properties in 2-normed spaces. In the case of bounded linear functionals on normed linear spaces the problem was first considered by R.R.Phelps [19]. For other related results see I. Singer’s book [20].

In the case of Banach spaces of Lipschitz functions similar results were obtained by the authors (see [1], [18]). The case of bilinear operators on 2 -normed spaces has been considered in [2].

Throughout this paper all the linear spaces will be considered over the field $K=\mathbf{R}$ or $K=\mathbf{C}$. A 2 -norm on a linear space $X$ of algebraic dimension at least 2 , is a functional $\|\cdot\|:,X\times X\rightarrow[0,\infty)$ verifying the axioms:

BN 1) $\|x,y\|=0$ if and only if $x,y$ are linearly dependent,
BN 2) $\|x,y\|=\|y,x\|$,
BN 3) $\|\lambda x,y\|=\|\lambda\|\cdot\|x,y\|$,

BN 4) $\|x+y,z\|\leq\|x,z\|+\|y,z\|$, for all $x,y,z\in X$ and $\lambda\in K$ (see [9])

If $\|\cdot,\cdot\|$ is a 2 -norm on the linear space $X$ then the function $\rho:X^{3}\rightarrow[0,\infty)$ defined by $\rho(x,y,z)=\|x-z,y-z\|,x,y,z\in X$ is a 2 -metric on $X$, in the sense of S.Gähler [8], which is translation invariant, i.e. $\rho(x+a,y+a,z+a)=\rho(x,y,z)$ for all $x,y,z\in X$ and a fixed element $a\in X$.

For a fixed $b\in X$, the function $p_{b}(x)=\|x,b\|,x\in X$, is a seminorm on $X$ and the family $P=\left\{p_{b}:b\in X\right\}$ of seminorms generates a locally convex topology on $X$, called the natural topology induced by the 2-norm $\|\cdot$,$\|.$

A pair $(X,\|\cdot\|$,$)whereX$ is a linear space and $\|\cdot$,$\|a2$-norm an $X$ will be called a 2 -normed space.

Remark 1. S.Gähler [10] considered only 2 -normed space over the field $\mathbf{R}$ of real numbers, but his definition automatically extends to the complex scalars too.

Let $(X,\|\cdot\|$,$)bea2$-normed space and $X_{1},X_{2}$ two subspaces of $X$. A 2 functional is an application $f:X_{1}\times X_{2}\rightarrow K$. The 2-functional $f$ is called bilinear if:

BL 1) $f\left(x+x^{\prime},y+y^{\prime}\right)=f(x,y)+f\left(x,y^{\prime}\right)+f\left(x^{\prime},y\right)+f\left(x^{\prime},y^{\prime}\right)$
BL 2) $f(\alpha x,\beta y)=\alpha\beta f(x,y)$,
for all $(x,y),\left(x^{\prime},y^{\prime}\right)$ in $X_{1}\times X_{2}$ and all $\alpha,\beta\in K$.
A 2-functional $f:X_{1}\times X_{2}\rightarrow K$ is called bounded if there exists a real number $L\geq 0$ (called a Lipschitz constant for $f$ ) such that

for all $(x,y)\in X_{1}\times X_{2}$.
This notion of boundedness was introduced by A.G.White Jr. [20] who defined also the norm of a bounded bilinear functional by:

Some immediate consequences of the definition are given in:
Proposition 2.1. (A.G. White Jr. [21].) Let ( $X,\|\cdot$,$\|)bea2-normedspace,X_{1},X_{2}$ two linear subspaces of $X$ and $f:X_{1}\times X_{2}\rightarrow K$ a bounded bilinear functional. Then
a) $f(x,y)=0$, for any pair $(x,y)\in X_{1}\times X_{2}$ of linear dependent elements;
b) $f(y,x)=-f(x,y)$, i.e. $f$ is an alternate bilinear functional;
c) The norm $\|f\|$ of $f$ can be calculated also by the formulae:

A.G.White Jr. [21] defined a kind of continuity for 2 -functionals, called subsequently 2 -continuity by S.Gähler [11].

A 2 -functional $f:X_{1}\times X_{2}\rightarrow K$, where $X_{1},X_{2}$ are linear subspaces of a 2 normed space $(X,\|\cdot\|$,$)iscalled2-continuousat\left(x_{0},y_{0}\right)\in X_{1}\times X_{2}$ if for every $\varepsilon>0$ there exists $\delta>0$ such that $\left|f(x,y)-f\left(x_{0},y_{0}\right)\right|<\varepsilon$ whenever
(i) $\left\|x,y-y_{0}\right\|<\delta$ and $\left\|x_{0}-x,y\right\|<\delta$, or
(ii) $\left\|x_{0}-x,y\right\|<\delta$ and $\left\|x_{0},y_{0}-y\right\|<\delta$

A 2-functional $f$ is called 2-continuous on $X_{1}\times X_{2}$ if it is 2-continuous at every point $(x,y)\in X_{1}\times X_{2}$.

An example of 2 -continuous 2 -functional is given by:

Proposition 2.2. (A.G. White Jr. [21, Th 2.2]) If ( $X,\|\cdot$,$\|)isa2-normedspacethen$ the 2 -functional $\|\cdot$,$\|is2$-continuous on $X\times X$.

It turns out that for bilinear functionals, boundedness and 2-continuity are equivalent and 2-continuity at $(0,0)$ implies 2-continuity on whole $X_{1}\times X_{2}$ :

Theorem 2.3. (A.G.White Jr. [21, Theorems 2.3 and 2.4]) a) A bilinear functional $f:X_{1}\times X_{2}\rightarrow K$ is 2-continuous on $X_{1}\times X_{2}$ if and only if it is bounded;
b) A bilinear functional $f:X_{1}\times X_{2}\rightarrow K$ which is 2-continuous at ( 0,0 ) is continuous on $X_{1}\times X_{2}$.
S.Gähler [11] remarked that 2-continuity of a 2-functional $f$ on $X\times X$ and its continuity with respect to the product topology on $X\times X$ are different notions. By proposition 2.2 a 2 -norm is a 2 -continuous functional on $X\times X$, but S.Gähler [11] exhibited an example of a 2 -norm which is not continuous on $X\times X$ (with respect to the product topology) and gave conditions ensuring the continuity of a 2 -norm on $X\times X$.

There are also examples of 2-functionals which are continuous on $X\times X$ with respect to the product topology but are not 2-continuous (see also S.Gähler [11]).

Let $(X,\|\cdot,\cdot\|)$ be a 2 -normed space, $X_{1},X_{2}$ two linear subspaces of $X$ and $f:X_{1}\times X_{2}\rightarrow K$ a bounded bilinear functional. The extension problem for $f$ consists in finding a bounded bilinear functional $F:X\times X\rightarrow K$ such that

We agree to call such an $F$ a norm preserving extension or a Hahn-Banach extension of $f$. As it was remarked by S.Gähler [11], p. 345 Korollar zu S. 5 und S.6, the norm preserving extension is not always possible. Some Hahn-Banach and Hahn type extension theorems for subspaces of the form $Y\times[b]$, where $Y$ is a linear subspace of $X$, $b\in X$ and $[b]$ denotes the subspace of $X$ spanned by $b$, were proved in the case of real 2 -normed spaces by A.G.White Jr. [21], S.Mabizela [17] and I.Franić [7].

In the following we shall show that all these extension results can be derived directly from the classical Hahn-Banach theorem. This approach allows to consider simultaneously both the cases of real and complex scalars.

Our methods of proofs rely upon slight extensions of Hahn-Banach and Hahn theorems from normed to seminormed spaces.

In what follows ( $X,p$ ) will denote a seminormed space (over the field $K=\mathbf{R}$ or C), with $p$ a nontrivial seminorm on $X$ (i.e. $p\neq 0$ ). It is well known that a linear functional $x^{*}$ is continuous on $X$ if and only if it is bounded (or Lipschitz) on $X$, i.e. there exists a number $L\geq 0$ such that

A number $L\geq 0$ verifying (3.2) is called a Lipschitz constant for $x^{*}$.
Proposition 3.1. Let ( $X,p$ ) be a seminormed space, $X^{*}$ its conjugate space and let $q:X^{*}\rightarrow[0,\infty)$ be defined by

Then
a) $\left|x^{*}(x)\right|\leq q\left(x^{*}\right)\cdot p(x)$, for all $x\in X$;
b) $q\left(x^{*}\right)=\inf\left\{L\geq 0:L\right.$ is a Lipschitz constant for $\left.x^{*}\right\}$;
c) The functional $q$ is a norm on $X^{*}$ and $\left(X^{*},q\right)$ is a Banach space.

Proof. a) Since $x^{*}\in X^{*}$ there exists $L\geq 0$ such that (3.2) holds. Now, if $x\in X$ is such that $p(x)=0$ then $x^{*}(x)=0$ too, and the inequality a) is trivially verified. If $p(x)>0$ then $p\left(\frac{1}{p(x)}\cdot x\right)=1$ so that $\left|x^{*}\left(\frac{1}{p(x)}\cdot x\right)\right|\leq q\left(x^{*}\right)$, which is equivalent to a).
b) If $L\geq 0$ verifies (3.2) then $\left|x^{*}(x)\right|\leq L$, for all $x\in X$ with $p(x)\leq 1$, implying $q\left(x^{*}\right)\leq L$. Since $L\geq 0$ is an arbitrary Lipschitz constant it follows

Because $q\left(x^{*}\right)$ is a Lipschitz constant for $x^{*}$ it follows that

implying the equality b).
c) It is immediate from (3.3) that $q$ is a seminorm on $X^{*}$. If $x^{*}\neq 0$ and $x_{0}\in X$ is such that $x^{*}\left(x_{0}\right)\neq 0$ then by a)

implying $q\left(x^{*}\right)>0$ and showing that $q$ is a norm on $X^{*}$.
The proof that $\left(X^{*},q\right)$ is a Banach space is standard and we omit it.

Theorem 3.2. (Hahn-Banach Theorem). Let ( $X,p$ ) be a seminormed space (over $K=\mathbf{R}$ or $\mathbf{C}$ ) with $p\neq 0,Y$ a linear subspace and $y^{*}\in Y^{*}$ a continuons linear functional on $Y$. Define $q_{1}\left(y^{*}\right)$ by

Then there exists a continuous linear functional $x^{*}$ on $X$ such that

where $q\left(x^{*}\right)$ is defined by (3.3).

Proof. The functional $p_{1}:X\rightarrow[0,\infty)$ defined by $p_{1}(x)=q_{1}\left(y^{*}\right)\cdot p(x),x\in X$ is a seminorm on $X$ and $\left|x^{*}(y)\right|\leq p_{1}(y)$ for all $y\in Y$, i.e. $y^{*}$ is dominated by $p_{1}$. By the Hahn-Banach Theorem (see e.g. [6] or [14]) there exists $x^{*}\in X^{*}$ such that

i) $\left.\quad x^{*}\right|_{Y}=y^{*}$
ii) $\left|x^{*}(x)\right|\leq q_{1}\left(y^{*}\right)\cdot p(x)$, for all $x\in X$.

Report issue for preceding element

By (3.6) ii) and Proposition 3.1 b) we obtain $q\left(x^{*}\right)\leq q_{1}\left(y^{*}\right)$. The reverse inequality follows from

$\square$

Hahn’s theorem ([6, Lemma II. 3.12) can be transposed to the seminormed case too

Theorem 3.3. (Hahn Theorem). Let ( $X,p$ ) be a seminormed space, $Y$ a linear subspace of $X$ and $x_{0}\in X\backslash\bar{Y}$. Then there exists a functional $x^{*}\in X^{*}$ such that

where $\delta=\inf\left\{p\left(x_{0}-y\right):y\in Y\right\}$.
Proof. Observe that $x_{0}\in X\backslash\bar{Y}$ implies $\delta>0$. Let $Z=Y\dot{+}Kx_{0}$ and let $z^{*}:Z\rightarrow K$ be defined by $z^{*}\left(y+\alpha x_{0}\right)=\alpha$, for $y\in Y$ and $\alpha\in K$. Obviously that $z^{*}$ is linear and, for $\alpha\neq 0$,

Since, for $\alpha=0,\left|z^{*}(y)\right|=0\leq\delta^{-1}\cdot p(y)$ it follows the continuity of $z^{*}$ and $q_{1}\left(z^{*}\right)\leq\delta^{-1}$, where $q_{1}\left(z^{*}\right)=\sup\left\{\left|z^{*}(z)\right|:z\in Z,p(z)\leq 1\right\}$. Taking a minimizing sequence $\left(y_{n}\right)\subseteq Y$ (i.e. $p\left(x_{0}-y_{n}\right)\rightarrow\delta$, for $n\rightarrow\infty$ ), we obtain

which for $n\rightarrow\infty$ gives $q_{1}\left(z^{*}\right)\geq\delta^{-1}$, implying $q_{1}\left(z^{*}\right)=\delta^{-1}$.
Now Theorem 3.3 follows from Theorem 3.2 applied to $Z$ and $z^{*}$. $\square$

Remark 2. The functional $x_{1}^{*}\in X^{*},x_{1}^{*}=\delta\cdot x^{*}$, verifies the conditions:

Pass now to the extension theorems for bounded bilinear functionals. The reduction to Hahn-Banach and Hahn’s theorems for bounded linear functionals on seminormed linear spaces will be based on the following result:

Proposition 3.4. Let ( $X,\|\cdot,\cdot\|$ ) be a 2-normed space (over $K=\mathbf{R}$ or $\mathbf{C}$ ), $Z$ a subspace of $X,b\in X\backslash\{0\}$ and let $[b]$ be the subspace of $X$ spanned by $b$. Denote by $p_{b}$ the seminorm on $Z$ given by

and let $q_{b}$ be its conjugate norm on $Z^{*}$, in the sense of Proposition 3.1. Then
a) If $f:Z\times[b]\rightarrow K$ is a bounded bilinear functional then the functional $z^{*}:Z\rightarrow K$ defined by $z^{*}(z)=f(z,b),z\in Z$ is a continuous linear functional on $Z$ and

b) Conversely, if $z^{*}$ is a bounded linear functional on $Z$, then the 2-functional $f:Z\times[b]\rightarrow K$ defined by $f(z,\alpha b)=\alpha z^{*}(z)$, for $(z,\alpha)\in Z\times K$, is a bounded bilinear functional and

Proof. a) Obviously that, for a given bounded bilinear functional $f:Z\times[b]\rightarrow K$, the functional $z^{*}:Z\rightarrow K$ defined by $z^{*}(z)=f(z,b),z\in Z$, is a linear functional on $Z$ and

for all $z\in Z$, implying that $z^{*}$ is a continuous linear functional on the seminormed space $\left(Z,p_{b}\right)$ and

On the other hand

implying that $q_{b}\left(z^{*}\right)$ is a Lipschitz constant for $f$, so that $\|f\|\leq q_{b}\left(z^{*}\right)$ and, therefore, $\|f\|=q_{b}\left(z^{*}\right)$.
b) Suppose now that $z^{*}$ is a given continuous linear functional on the seminormed space $\left(Z,p_{b}\right)$ and define $f:Z\times[b]\rightarrow K$ by $f(z,\alpha b)=\alpha\cdot z^{*}(z),(z,\alpha)\in Z\times K$. Obviously that $f$ is a bilinear functional and

for all $(z,\alpha)\in Z\times K$, showing that $f$ is a bounded bilinear functional and that $\|f\|\leq q_{b}\left(z^{*}\right)$.

Taking into account the fact that $p_{b}(z)=\|z,b\|$ we obtain

Again the equality $\|f\|=q_{b}\left(z^{*}\right)$ holds.

Now we are in position to prove the promised extension theorem.

Theorem 3.5. (Hahn-Banach Extension Theorem, A.G.White Jr. [21, Th.2.7]) Let $(X,\|\cdot\|$,$)bea2$-normed space (over $K=\mathbf{R}$ or $\mathbf{C}$ ), $Y$ a subspace of $X,b\in X$ and let [b] be the subspace of $X$ spanned by b. If $f:Y\times[b]\rightarrow K$ is a bounded bilinear functional then there exists a bounded bilinear functional $F:X\times[b]\rightarrow K$ such that

Proof. Let $p_{b}:X\rightarrow[0,\infty)$ be the seminorm defined by $p_{b}(x)=\|x,b\|,x\in X$, and let $y^{*}:Y\rightarrow K$ be given by $y^{*}(y)=f(y,b)$. Then by Proposition 3.4 a ), $y^{*}$ is a continuous linear functional on $Y$ and $q_{b}^{\prime}\left(y^{*}\right)=\|f\|$, where

By Theorem 3.2 there exists a bounded linear functional $x^{*}\in X^{*}$ such that $\left.x^{*}\right|_{Y}=y^{*}$ and $q_{b}\left(x^{*}\right)=q_{b}^{\prime}\left(y^{*}\right)$, where

Defining now $F:X\times[b]\rightarrow K$ by $F(x,\alpha b)=\alpha\cdot x^{*}(x)$, for $(x,\alpha)\in X\times K$ and applying Proposition 3.4 b) it follows that the bilinear functional $F$ fulfils all the requierements of the Theorem. $\square$

The analogue of Hahn’s theorem for bilinear functionals is:

Theorem 3.6. (S.Mabizela [17, Th.2]) Let ( $X,\|\cdot$,$\|)bea2$-normed space over $K=\mathbf{R}$ or $\mathbf{C},Y$ a linear subspace of $X,b\in X$ and $[b]$ the subspace of $X$ spanned by $b$. If $x_{0}\in X$ is such that $\delta>0$, where

then there exists a bounded bilinear functional $F:X\times[b]\rightarrow K$ such that

Proof. Consider again the seminormed space $\left(X,p_{b}\right)$, where $p_{b}(x)=\|x,b\|,x\in X$, and apply Theorem 3.3 to obtain a bounded linear functional $x^{*}$ on $X$ such that

where $q_{b}\left(x^{*}\right)$ is given by (3.11).
Defining $F:X\times[b]\rightarrow K$ by $F(x,\alpha b)=\alpha\cdot x^{*}(x),(x,\alpha)\in X\times K$, and applying Proposition 3.4 b), it follows that the bounded bilinear functional $F$ verifies the conditions (3.13) of the Theorem. $\square$

Remark 3. S.Mabizela [17, Th.2] requieres for $x_{0}$ and $b$ to be linearly independent. Observe that if $x_{0},b$ are linearly dependent then, by the axiom

BN 1) in Section 1, $\left\|x_{0},b\right\|=0$ and a fortiori $\delta=0$, because

Therefore the hypothesis $\delta>0$ forces $x_{0}$ and $b$ to be linearly independent and $x_{0}\in X\backslash\bar{Y}$, where $\bar{Y}$ denotes the closure of $Y$ in the seminormed space $\left(X,p_{b}\right)$.

An immediate consequence of Theorem 3.6 is the following result, known also as Hahn’s Theorem:

Theorem 3.7. If ( $X,\|\cdot$,$\|)isa2$-normed space and $x_{0},b$ are linearly independent elements in $X$ then there exists a bounded bilinear functional $F:X\times[b]\rightarrow K$ such that:

Proof. Putting $Y=\{0\}$ in Theorem 3.6 and taking into account the linear independence of $x_{0}$ and $b$, one obtains $\delta=\left\|x_{0},b\right\|>\dot{0}$.

By Theorem 3.6, it follows the existence of a bounded bilinear functional $G$ : $X\times[b]\rightarrow K$ such that $G\left(x_{0},b\right)=1$ and $\|G\|=\delta^{-1}$. Then $F=\delta\cdot G$ satisfies the conditions (3.15) of the theorem.

For a 2 -normed space $(X,\|\cdot\|$,$),asubspaceY$ of $X$ and $b\in X$ denote by $Y_{b}^{\sharp}$ the linear space of all bounded bilinear functionals on $Y\times[b]$. Equipped with the norm (2.2), $Y_{b}^{\sharp}$ is a Banach space (see A.G.White Jr.[20]) The Banach space $X_{b}^{\sharp}$ is defined similarly.

For $f\in Y_{b}^{\sharp}$ denote by $E(f)$ the set of all norm-preserving extensions of $f$ to $X\times[b]$, i.e.

By Theorem 3.5, $E(f)\neq\phi$ and $E(f)$ is a convex subset of the unit sphere $S(0,\|f\|)=\left\{G\in X_{b}^{\sharp}:\|G\|=\|f\|\right\}$. Indeed, for $F_{1},F_{2}\in E(f)$ and $\lambda\in[0,1]$,

and

Denoting $G=\lambda F_{1}+(1-\lambda)F_{2}$ it follows $\left.G\right|_{Y\times[b]}=f$ and

For a subspace $Y$ of a 2 -normed space $(X,\|\cdot\|$,$)let$

be the annihilator of $Y$ in $X_{b}^{\sharp}$.
For a nonvoid subset $Z$ of $X_{b}^{\sharp}$ the distance of an element $F\in X_{b}^{\sharp}$ to $Z$ is defined by

An element $G_{0}\in Z$ such that $\left\|F-G_{0}\right\|=d(F,Z)$ is called an element of best approximation (or a nearest point) for $F$ in $Z$.

Let

denote the set of all elements of best approximation for $F$ in $Z$. The set $Z$ is called proximinal if $P_{Z}(F)\neq\emptyset$ for all $F\in X_{b}^{\sharp}$, Chebyshev provided $P_{Z}(F)$ is a singleton for all $F\in X_{b}^{\sharp}$ and semi-Chebyshev if $\operatorname{card}P_{Z}(F)\leq 1$, for all $F\in X_{b}^{\sharp}$.

A subspace of the form $Y_{b}^{\perp}$ of $X_{b}^{\sharp}$ is always proximinal and we have simple formulae for the distance of an element $F\in X_{b}^{\sharp}$ to $Y_{b}^{\perp}$ and for the set of nearest points.

Theorem 4.1. If $(X,\|\cdot\cdot\|)$ is a 2 -normed space, $Y$ a subspace of $X,b\in X$ and $F\in X_{b}^{\#}$ then

Moreover, $Y_{b}^{\perp}$ is a proximinal subspace of $X_{b}^{\#}$ and

Proof. Since $\left.(F-G)\right|_{Y\times[b]}=\left.F\right|_{Y\times[b]}$, for any $G\in Y_{b}^{\perp}$ it follows

so that

To prove the reverse inequality observe that $f=\left.F\right|_{Y\times[b]}\in Y_{b}^{\sharp}$. Now if $H$ is a normpreserving extension of $f$ to $X\times[b]$ then $F-H\in Y_{b}^{\perp}$ and

proving the formula (4.5).
For $H\in E\left(\left.F\right|_{Y\times[b]}\right)$ we have $F-H\in Y_{b}^{\perp}$ and $\|F-(F-H)\|=\|H\|=\left\|\left.F\right|_{Y\times[b]}\right\|=d\left(F,Y_{b}^{\perp}\right)$, showing that $F-H$ is a nearest point to $F$ in $Y^{\perp}$.

Conversely, if $G$ is a nearest point to $F$ in $Y_{b}^{\perp}$ then $\left.(F-G)\right|_{Y\times[b]}=\left.F\right|_{Y\times[b]}$ and, denoting $H=F-G$, it follows $G=F-H$ and

showing that $H$ is a norm preserving extension for $\left.F\right|_{Y\times[b]}$. The equality (4.6) is proved and since, by Theorem 3.5, $E\left(\left.F\right|_{Y\times[b]}\right)\neq\emptyset$, for all $F\in X_{b}^{\sharp}$, it follows the proximinality of the subspace $Y_{b}^{\perp}$ in $X_{b}^{\sharp}$.

Now we are in position to state and prove the duality theorem relating the uniqueness of extension and of best approximation. Recall that for normed linear spaces and bounded linear functionals a similar result was first proved by R.R.Phelps [18].

Theorem 4.2. Let $(X,\|\cdot\|$,$)bea2-normedspace,Y$ a subspace of $X$ and $b\in X$. Then the following assertions are equivalent:
$1^{0}$ Every $f\in Y_{b}^{\sharp}$ has a unique norm preserving extension to $X\times[b]$;
$2^{0}Y_{b}^{\perp}$ is a Chebyshev subspace of the Banach space $X_{b}^{\sharp}$.
Proof. The Theorem is an immediate consequence of the formula (4.6) from Theorem 4.1.

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Babes-Bolyai University, Faculty of Mathematics and Computer Science, Str. M. Kogălniceanu 1, RO-3400 Cluj-Napoca, Romania.

Institute of Mathematics, 37, Republicii Str., Cluj-Napoca, Romania

All groups considered in the paper are finite. We denote by $o$ an arbitrary set of primes and by $o^{\prime}$ the complement to $o$ in the set of all primes.

Definition 1.1. a) A class $\underline{X}$ of groups is a homomorph if $\underline{X}$ is closed under homomorphisms.
b) A group $G$ is primitive if $G$ has a stabilizer, i.e. a maximal subgroup $W$ with $\operatorname{core}_{G}W=1$, where

c) A homomorph $\mathbf{X}$ is a Schunck class if $\mathbf{X}$ is primitively closed, i.e. if any group $G$, all of whose primitive factor groups are in $\underline{\mathbf{X}}$, is itself in $\underline{\mathbf{X}}$.