Derivations on Group Algebras with Coding Theory Applications

This paper classifies the derivations of group algebras in terms of the generators and defining relations of the group. If $RG$ is a group ring, where $R$ is commutative and $S$ is a set of generators of $G$ then necessary and sufficient conditions on a map from $S$ to $RG$ are established, such that the map can be extended to an $R$-derivation of $RG$. Derivations are shown to be trivial for semisimple group algebras of abelian groups. The derivations of finite group algebras are constructed and listed in the commutative case and in the case of dihedral groups. In the dihedral case, the inner derivations are also classified. Lastly, these results are applied to construct well known binary codes as images of derivations of group algebras.


Introduction
Group rings and derivations of rings have both been studied for more than 60 years. For a history of group rings see Polcino Milies and Sehgal [16] and for a survey article on derivations see Ashraf, Ali, and Haetinger [2]. The results of Posner [17] and Herstein [6] attracted particular attention. Prime, semiprime and 2-torsion free rings were a focus of the resulting research.
Derivations of C * -algebras have been studied by several authors. In [19], Sakai proved that every derivation of a simple C * -algebra becomes inner in its multiplier algebra. Mathieu and Villena, in [14] study the structure of Lie derivations on C * -algebras. In the 2000 paper Derivations on Group Algebras [5], Ghahramani, Runde and Willis, examine the first cohomology space of the group algebra L 1 (G), where G is a locally compact group. The derivation problem asks whether every derivation from L 1 (G) to M (G) is inner, where G is a locally compact group and M (G) is the multiplier algebra of L 1 (G). It was solved by Losert [12]. The 2017 preprint "Derivations of Group Algebras", [1] by Arutyunov, Mishchenko and Shtern describes the outer derivations of L 1 (G).
Group rings have been used to construct new codes as well as to study existing codes. In [10] Hurley and Hurley present techniques for constructing codes from group rings. The codes constructed consist primarily of two types, zero-divor codes and unit-derived codes. The structure of group ring codes is examined in [9]. The author gives a decomposition of a group ring code into twisted group ring codes and proves the nonexistence of self-dual group ring codes in particular cases.
Derivations have also been employed in coding theory. In [3] codes are constructed as modules over skew polynomial rings, where the multiplication is defined by a derivation and an automorphism. In this paper we are primarily concerned with derivations of group algebras and their application to coding theory.
However, there has not been as much research into derivations of group algebras with positive characteristic. Notable exceptions include Smith [20], Spiegel [21] and Ferrero, Giambruno and Polcino Milies [13]. In the latter paper the authors prove the following theorem.
Theorem 1.1. [13] Let R be a semiprime ring and G a torsion group such that [G ∶ Z(G)] < ∞, where Z(G) denotes the center of G. Suppose that either char R = 0 or for every characteristic p of R, p ffl o(g), for all g ∈ G. Then every R-derivation of RG is inner.
In this paper we are particularly interested in finite group algebras. This is motivated in part by applications to error correcting codes. Theorems 1.1 and 3.1 direct our focus, in the commutative case, to the study of derivations of modular (nonsemisimple) group algebras with positive characteristic. Theorem 2.2 shows that when K is an algebraic extension of a prime field all derivations of a K-algebra are K-derivations. If RG is a group ring, where R is commutative and S is a finite set of generators of G then necessary and sufficient conditions on a map f ∶ S → RG are established, in Theorem 2.5, such that f can be extended to an R-derivation of RG. Section 3 outlines some applications of the results of Section 2. All derivations of finite commutative group algebras of positive characteristic are determined in Theorem 3.4. If G is a finite abelian group and K a finite field of positive characteristic p then the image of a minimum set of generators of the Sylow p-subgroup of G under a derivation of KG can be chosen arbitrarily, however this is not always the case in the noncommutative setting. An inner derivation of a ring R maps a ∈ R to ab − ba, for some element b ∈ R. In the case of finite dihedral group algebras a basis is given for the space of derivations in Theorem 3.11 and also for those that are inner in Theorem 3.13.
The extended binary Golay [24, 12,8] code and the extended binary quadratic residue [48,24,12] code are both presented as images of derivations of group algebras in Section 3.3. Definition 1.2. Notation: N, Z and Q denote the natural numbers, the integers and the rational numbers, and F p n denotes the finite field with p n elements. The group ring RG denotes the set of all formal linear combinations of the form ∑ g∈G a g g, of finite support where a g ∈ R, together with the operations of addition (componentwise) and multiplication defined as (∑ g∈G a g g)(∑ h∈G b h h) = ∑ g,h∈G a g b h gh. We adopt the usual convention that empty sums are 0 and empty products are 1.

Derivations of Group Rings
In this section we establish necessary and sufficient conditions on a map f ∶ S → RG, such that f can be extended to an R-derivation of the group ring RG, where S is a finite set of generators of G and R is commutative. First, some identities and preliminary results are presented.
, for all a ∈ R which commute with d(a) and k ∈ N.
for all units a ∈ R which commute with d(a) and k ∈ Z.
Proof. (i) We will prove Equation 3 by induction on m. Base case: m = 1. This is true as (iv) Let a be an element of R that commutes with d(a). Then using Equation 4 (v) Let a be a unit which commutes with d(a). Then a −1 is also a unit which commutes with Furthermore, 0 = d(1) = d(a 0 ) = 0a −1 d(a) and so Equation (7) holds for all integers k.
The following Theorem shows that when K is an algebraic extension of a prime field all derivations of a K-algebra are K-derivations.
Theorem 2.2. Let A be a K-algebra where K is an algebraic extension of a prime field F and let d ∈ Der(A). Then d(K) = {0} and d is a K-linear map.
Let a be a nonzero element of K and let m a (x) = ∑ na j=0 b aj x j ∈ F [x] be the minimal polynomial of a over F . a is a central unit in K and so Equation 7 of Lemma 2.1 applies. Note that for b ∈ F and α ∈ K we have d(bα) = bd(α), since d(F ) = 0. Thus applying a derivation d to m a (a) = 0 and using Equation 7 where q is a polynomial in F [x]. Moreover, q(a) ≠ 0 as this would contradict the minimality of the degree of m a (x). Therefore d(a) = 0, since q(a) is invertible as it is a non zero element of the field K. Hence d(K) = {0}.
The K-linearity of d is immediate since d is additive and if a ∈ A and k ∈ K then d(ka) = d(k)a + kd(a) = 0 + kd(a). Corollary 2.3. Let K be an algebraic extension of a prime field F . Let G be a torsion group such that [G ∶ Z(G)] < ∞, where Z(G) denotes the center of G. Suppose that either char(K) = 0 or that char(K) = p > 0, and p does not divide the order of g, for all g ∈ G. Then every derivation of KG is inner.
Proof. By Theorem 2.2, every derivation of KG is a K-derivation and since every field is semiprime, Theorem 1.1 implies that every derivation of KG is inner.
Note that the requirement that K is algebraic over F is necessary in Theorem 2.2 as the following example shows.
Example 2.4. Let Q(t) be a transcendental extension of the rationals (the field of rational functions of t). Since Q(t) is a Q-algebra, Theorem 2.2 implies that d(Q) = {0} for all derivations d of Q(t). However, by Proposition 5.2 of Chapter VIII in [11], there exists a nonzero derivation d of Q(t), since Q(t) is a finitely generated extension over Q that is not separable algebraic.
Theorem 2.5. Let G = ⟨S T ⟩ be a finitely generated group, where S is a finite generating set and T a set of relators. Let F S be the free group on S and φ∶ F S → G the homomorphism of F S onto G. Let R be a commutative unital ring and f a map from S to RG. Then (i) f can be uniquely extended to a map f * from F S to RG such that (iii) if f can be extended to an R-derivation of RG, then this extension is unique.
Proof. Let f be a map from S to RG. φ is the identity map on S, so for Define f * ∶ F S → RG as follows: Let 0 ≤ l ≤ k and u = ∏ l i=1 w i and v = ∏ k i=l+1 w i . Then by Equations 9 and 10 Therefore f * defined by Equations 9 and 10 satisfies Equation 8.
If w is a word on S, denote the reduced word by w.
In order for f * to be well defined on F S we need to show that f * (w) = f * (w) for all words w on S. Let u, v be words on S and let a ∈ S.
Then by Equation 10, f * (aa −1 ) = 0 and so by Equation Therefore f * (w) = f * (w) for all words w on S. We now prove the uniqueness of f * .
Assume that there exists a map f * ∶ F S → RG, distinct from f * which is also an extension of f and which also satisfies Equation 8. Let 1 be the identity element of F S . Then x i and x c are both elements of F S whose length is less than c. Therefore by Equation 8 (ii) Considering S as a subset of G, suppose that the map f ∶ S → RG can be extended to an Then by Equations 10 and 3 This proves the implication in (ii).
Let N = ⟨T F S ⟩ be the smallest normal subgroup of F S containing T . Any non-identity element n of N can be written as . Therefore by Equations 10 and 11 Also φ(n) = 1, for all n ∈ N and so for any . Thereforẽ f (g) =f (g) for all g ∈ G such that l(g) < 2 and so x can be written as x = yz, where y, z ∈ G such that l(y) < l(x) and l(z) < l(x). Thenf (x) =f (yz) =f (y)z+yf (z) =f (y)z+yf (z) =f (x). This contradiction implies thatf is the unique extension of f such thatf (gh) =f (g)h + gf (h) for any g, h ∈ G. Extendf , R-linearly to RG and denote this unique extension also byf . Let asf is an R-linear map. Moreover Therefore the mapf obeys Leibniz's rule for all products of elements of RG and so is an R-derivation of RG. This proves (ii) and (iii).
Corollary 2.6. Let G = ⟨S T ⟩ be a finitely generated group, where S is a finite generating set and T a set of relators. Let F S be the free group on S and φ∶ F S → G the homomorphism of F S onto G. Let K be an algebraic extension of a prime field and f a map from S to KG. Then Remark 2.7. The restriction that R be a commutative ring in Theorem 2.5 is necessary. To demonstrate this, let r 1 , r 2 be noncommuting elements in a ring R and let G be the infinite cyclic group generated by S = {s}, that is the free group on S. Let f ∶ S → RG be the map defined by s ↦ r 1 and extend f to a map f * ∶ G → RG as in Theorem 2.5 (i). Assume that f can be extended to an R-derivation d of RG. Then d(s)r 2 s + sd(r 2 s) = r 1 r 2 s + sr 2 d(s) = r 1 r 2 s + sr 2 r 1 = (r 1 r 2 + r 2 r 1 )s.
Therefore the Leibniz rule does not apply since d(sr 2 s) ≠ d(s)r 2 s + sd(r 2 s). This contradicts the assumption that f can be extended to an R-derivation of RG.

Applications
We will now apply the results of the previous sections to finite commutative group algebras in Section 3.1 and then to finite dihedral group algebras in Section 3.

Derivations of Commutative Group Algebras
The next result directs our study of derivations of commutative group algebras to the nonsemisimple case.  Note that (i) of this Corollary also follows from Theorem 1.1.
Therefore f * ([x k , x l ]) = 0, for all k, l = 1, 2, . . . , n. Also by Equation 10 since KG has characteristic p. Therefore any map f ∶ S → KG can be uniquely extended to a derivation of KG. By Lemma 1.5 Der(KG) is a vector space over K.
The extension of f to a derivation of KG is ∑ n i=1 ∑ g∈G k i,g gB i . Therefore any derivation of KG can be written as a K-linear combination of the elements of B. Furthermore, if (∑ n i=1 ∑ g∈G k i,g gB i )(x j ) = 0, then ∑ g∈G k g,j g = 0, which implies k g,j = 0 for all g ∈ G. Therefore the elements of B are Klinearly independent and so form a basis of Der(KG). As we will see in the next section, derivations of noncommutative finite group algebras are more involved.

Derivations of Dihedral Group Algebras
Let n be an integer greater than 2 and let D 2n denote the dihedral group with 2n elements and presentation ⟨x, y x n = y 2 = (xy) 2 = 1⟩. This section classifies the derivations of the group algebra F 2 m D 2n . Definition 3.6. Let RG be a group ring. The augmentation ideal of RG, denoted by ∆(G), is the kernel of the homomorphism from RG to R defined by ∑ g∈G a g g ↦ ∑ g∈G a g .  Corollary 3.9. Let C(y) and C(xy) denote respectively the centralisers of y and xy in F 2 m D 2n . Then the following are bases for C(y) and C(xy). Case (1): n is even Case (2): n is odd Proof. Let g ∈ D 2n and denote by Orb(g y ) the subset {g, g y } of D 2n . The set {Orb(g y ) g ∈ G} is a partition of D 2n . The set of elements formed by taking the partition sums forms a basis B e (y) for C(y), when n is even and B o (y), when n is odd. The map α∶ D 2n → D 2n defined by y ↦ xy and x ↦ x is an automorphism of D 2n . Extend α F 2 m -linearly to an F 2 m -algebra automorphism of F 2 m D 2n .
This implies a = a y and b = b y and so c ∈ C(y). Therefore c ∈ C(y) if and only if α(c) ∈ C(xy). Applying α to the basis B e (y) gives B e (xy) and applying α to B o (y) gives B o (xy).
and y ′ = d(y). Note that d(x) and d(y) uniquely determine this derivation.
By Lemma 1.5, Der(F 2 m D 2n ) forms a vector space over F 2 m . The following Theorem exhibits a basis for Der(F 2 m D 2n ).
can be extended to a derivation of F 2 m D 2n if and only if f (y) ∈ C(y) and f (x)y + xf (y) ∈ C(xy).
Therefore Der(F 2 m D 2n ) = {d (Λy+xΩy,Ω) }, where and Ω = r 1 1 + r 2 y + We now give a basis for the set of inner derivations of F 2 m D 2n .
The union of the disjoint sets B and B Z is a basis for F 2 m D 2n . Case(2) n is odd. Write n = 2c+1. By Lemma 3.8, Again, the disjoint union of B and B Z is a basis for F 2 m D 2n .
Therefore the set {d b b ∈ B} spans the set of inner derivations of F 2 m D 2n . Moreover, if ) which implies that a i = 0, for i = 1, 2, . . . , 3⌊ n−1 2 ⌋ and so the set {d b b ∈ B} forms a basis for the vector space of inner derivations of F 2 m D 2n .
The derivation problem asks whether every derivation from L 1 (G) to M (G) is inner, where G is a locally compact group and M (G) is the multiplier algebra of L 1 (G). It was solved by Losert [12]. We can ask a similar question for finite group algebras. Let KG be a group algebra where both K and G are finite. Are all derivations of KG inner? Theorems 3.11 and 3.13 show that the dimension of Der(F 2 m D 2n ) is greater than the dimension of the inner derivations of F 2 m D 2n and so not all derivations of F 2 m D 2n are inner. However does there exist an algebra A ⊃ KG such that all derivations of KG become inner in A? Theorem 3.15 answers this question.
where addition is performed componentwise and multiplication satisfies the relation xa = ax + δ(a), for all a ∈ R is called a differential polynomial ring.
Theorem 3.15. Let G be a finite group and KG be the group algebra over the finite field K.
. Therefore the restriction of D x to KG is equal to d.

Applications to Coding Theory
Example 3.16. Let C 24 = ⟨x x 24 = 1⟩ and let d∶ F 2 C 24 → F 2 C 24 be the derivation defined by x ↦ 1 + x + x 3 + x 4 + x 5 + x 7 + x 9 + x 12 (by Theorem 3.4 this uniquely defines a derivation). Then by Lemma 2.1, d(x 2n ) = 0 and d(x 2n+1 ) = x 2n d(x), for n ∈ {0, 1, . . . , 11}. The image of the group algebra under this derivation is a binary code of length 24 and dimension 12. A generator matrix G 24 of this code is given in Figure 1.
Again by Theorem 3.4 this uniquely defines a derivation of F 2 C 48 . The image of the group algebra under this derivation is a binary [48, 24, 12] doubly even self dual code (verified using GAP 4.8.6 [4]). It is equivalent to the extended binary quadratic residue code of length 48 [8]. A generator matrix for this code is given by the block matrix [I 24 A], where I 24 is the identity of the ring of 24 × 24 matrices over F 2 and A is the matrix given in Figure 2.   Figure 2: The right hand block of a generator matrix of the binary [48, 24, 12] code defined by the derivation δ.