Group homomorphisms as error correcting codes

We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups $G$ and $H$. We prove some general structural results on how the distance behaves with respect to natural group operations, such as passing to subgroups and quotients, and taking products. Our main result is a general formula for the distance when $G$ is solvable or $H$ is nilpotent, in terms of the normal subgroup structure of $G$ as well as the prime divisors of $|G|$ and $|H|$. In particular, we show that in the above case, the distance is independent of the subgroup structure of $H$. We complement this by showing that, in general, the distance depends on the subgroup structure $G$.


Error correcting codes
The theory of error correcting codes studies codes, which are subsets of Σ n for some alphabet Σ and block length n. The distance between two strings of equal length is the number of coordinates in which they differ. The distance ∆ of a code is simply the minimum distance between any pair of distinct codewords (elements of the code). Hamming [Ham50] identifies the distance of a code as the key parameter measuring the error correcting capability of the code. As long as the number of coordinates in which a codeword is corrupted is less than ∆/2, one can uniquely recover the original codeword. Elias [Eli57] and Wozencraft [Woz58] proposed list decoding, in which one insists only on recovering a list, whose size is at most polynomial in n, which contains the original codeword. The Johnson bound [Joh62] shows that codes can list decode errors beyond ∆/2. Codes with efficient list decoding algorithms include the Hadamard code [GL89], Reed-Solomon codes and variants thereof [Sud97,GS99,GR08,Gur11], Reed-Muller codes [GKZ08,Gop13], multiplicity/derivative codes [Kop12,GW11], and abelian group homomorphisms [GKS06,DGKS08]. For some of these codes, in particular for carefully chosen subcodes the folded Reed-Solomon codes and multiplicity/derivative codes [DL12], the Reed-Muller codes, and abelian group homomorphisms, it was shown that for any constant ǫ > 0 one can algorithmically list decode up to ∆ − ǫn errors with a constant list size, depending only on 1/ǫ. For all of these codes, the codewords are interpreted as certain functions f : A → B from some domain A to codomain B. In this case, the coordinates of the codeword are indexed by A and the alphabet is B.
In a companion work [GS14], the author and Sudan show the analogous list decoding results for group homomorphisms between supersolvable groups. A technical obstacle which did not arise in the previous works of [ GKS06,DGKS08] on list decoding abelian group homomorphisms is actually determining the distance of the code. This turns out to be a nontrivial problem and serves as the primary motivation of this paper.

Group homomorphisms
Let G and H be finite groups, with homomorphisms Hom(G, H). A function φ : G → H is a (left) affine homomorphism if there exists h ∈ H and φ 0 ∈ Hom(G, H) such that φ(g) = hφ 0 (g) for every g ∈ G. The set of left affine homomorphisms from G to H by aHom(G, H). Note that the set of left affine homomorphisms equals the set of right affine homomorphisms, since The equalizer of two functions f, g : G → H, denoted Eq(f, g), is the set More generally, if Φ ⊆ {f : G → H} is a collection of functions, then the equalizer of Φ is the set In the theory of error correcting codes, the usual measure of distance between two strings is the relative Hamming distance, which is the fraction of symbols on which they differ. In the context of group homomorphisms, we find it more convenient to study the complementary notion, the fractional agreement. We define the agreement agr(f, g) between two functions f, g : G → H to be the quantity The maximum agreement of the code aHom(G, H), denoted by Λ G,H , is defined as In Section 2, we study the structure of the equalizers of homomorphisms and prove some basic results that will be useful later. As we will see (Proposition 2.5), adding affine homomorphisms does not change the distance of this code. However, we include these functions in the code so that Λ G,H is well-defined when | Hom(G, H)| = 1, as long as H is nontrivial.

Our results
Our main result is the following formula for Λ G,H when G is solvable or H is nilpotent.
In Section 3, we prove general facts about Λ G,H , such as how it behaves with respect to group decompositions, subgroups, and quotients.
The proof of Theorem 1.1 is divided into two sections. Section 4 handles the case where H is nilpotent, and Section 5 handles the case where G is solvable.
In Section 6, we investigate Λ G,H when G is a non-abelian simple group, and in particular when G = A n is the alternating group on n ≥ 5 objects. We show that the formula for Λ G,H for solvable G does not apply to non-abelian simple groups, and hence does not extend to arbitrary groups. We also see that, in general, Λ G,H depends not only on the prime divisors of G and H but also on the subgroup structure of H, in particular whether H contains isomorphic copies of G and how these copies are embedded in H.

Equalizers
We begin by observing that the equalizer of a set of (affine) homomorphisms is a (coset of a) subgroup of G. A basic question we would like to answer is the following: if φ, ψ ∈ Hom(G, H), then must the index of Eq(φ, ψ) divide |H|? Note that this is true when one of the homomorphisms, say ψ, is the trivial homomorphism mapping to 1 H , so that Eq(φ, ψ) = ker φ. This follows from the fact that G/ ker φ ∼ = im φ which is a subgroup of H, so [G : ker φ] = | im φ| divides H. We will show in Proposition 2.6 that the more general statement holds when H is a p-group. Before doing so, we collect a few more basic facts that will be useful to us.
Proposition 2.2. Let G and H be finite groups and let Φ ⊆ Hom(G, H). For h ∈ H, if the set φ∈Φ φ −1 (h) is nonempty, then it is a coset of the subgroup φ∈Φ ker φ.
Proof. Consider the homomorphism φ : G → k i=1 (G/N i ) defined by φ(g) = (gN 1 , . . . , gN k ). Then ker φ = k i=1 N i = N , which shows that N is a normal subgroup. Moreover, im φ is a subgroup of k i=1 (G/N i ), and by the First Isomorphism Theorem, G/N = G/ ker φ ∼ = im φ.
Proposition 2.4. Let G and H be finite groups, and let Φ ⊆ Hom(G, H). Let K ⊆ H be the set of h ∈ H such that φ∈Φ φ −1 (h) is nonempty. Then Proof. We decompose Eq(Φ) into the disjoint union The result then follows from the fact that each φ∈Φ φ −1 (h) is a coset of φ∈Φ ker φ, which follows from Proposition 2.2.
The following proposition is simply the observation that the maximum agreement between two affine homomorphisms is achievable by two homomorphisms, which will allow us to reason about homomorphisms rather than affine homomorphisms in later proofs, without loss of generality.
Proposition 2.5. If G and H are finite groups, then there exist φ, ψ ∈ Hom(G, H) such that Finally, we conclude this section by proving the following.
Proposition 2.6. Let G be a finite group and let H be a finite p-group. If Φ ⊆ aHom(G, H) and Eq(Φ) = ∅, then [G : Eq(Φ)] is a power of p. In particular, Proof. By Proposition 2.1, we may assume that Φ ⊆ Hom(G, H). It follows from Proposition 2.3 that G/ φ∈Φ ker φ is isomorphic to a subgroup of φ∈Φ (G/ ker φ) ∼ = φ∈Φ im φ. But the im φ are subgroups of H, so they are p-groups, hence φ∈Φ im φ is a p-group, and so G/ φ∈Φ ker φ is a p-group, i.e. |G| φ∈Φ ker φ = p k for some k. By Proposition 2.4, there is some integer m such that By Proposition 2.1, Eq(Φ) is a subgroup of G, and so by Lagrange's theorem, p k m = |G| | Eq(Φ)| is an integer, hence m divides p k , therefore p k m is a power of p.

General facts
In this section, we investigate general properties of Λ G,H .

Subgroups and Quotients
Proposition 3.1. If G and H are finite groups and K ≤ H is a subgroup, then Proof. This follows from the fact that aHom(G, K) ⊆ aHom(G, H).
Proposition 3.2. If G, H are nontrivial finite groups and N ⊳ G is a normal subgroup, then This follows from the fact that φ and ψ are constant on cosets, so Eq(φ, ψ) is a disjoint union of cosets, and the cosets xN on which φ and ψ agree are exactly those for which φ G/N (xN ) = ψ G/N (xN ). Proof. Let φ ∈ Hom(G, H). The restriction of φ to the domain S is a homomorphism in Hom(S, H), which is trivial by assumption. This means that S ≤ ker φ. Since ker φ G, by minimality of N it follows that N ≤ ker φ. In particular, φ = φ ′ • π where φ ′ ∈ Hom(G/N, H) and π : G → G/N is the natural quotient map.

Key facts
Here we prove some key facts that will help us characterize Λ G,H when G is solvable.
Lemma 3.8. If G has a normal subgroup of index p and p divides |H|, then Proof. Let N ⊳ G be a normal subgroup of index p. Let φ 1 : G → G/N be the natural quotient homomorphism. Since p divides |H|, by Cauchy's theorem, there is an element h ∈ H of order p. The subgroup h ≤ H generated by h is isomorphic to Z p , and since G/N has order p, it is also isomorphic to Z p , hence there is an isomorphism φ 2 : G/N → h . Define φ : G → H to be the composition φ = φ 2 • φ 1 . Since φ 1 , φ 2 are homomorphisms, φ is a homomorphism, and moreover since φ 2 is an isomorphism, ker φ = ker φ 1 = N . Therefore, | ker φ| = |N | = |G|/p.

Nilpotent codomain
In this section, we prove Theorem 1.1 when H is nilpotent. We begin by considering the case where G has no normal subgroups of index p for any prime p dividing gcd(|G|, |H|). The following fact will be useful.
Proposition 4.1. If G is a finite solvable group and N ⊳ G is a maximal normal subgroup, then N has prime index in G.
We proceed to prove that Λ G,H = 0. In fact, we prove it for the case where H is solvable. This does not hold in general as, for instance, when G = H = A n for n ≥ 5, which is a nonabelian simple group, G has no normal subgroups of prime index, yet there are certainly nontrivial homomorphisms A n → A n . Now we proceed to the case where G has a normal subgroup of index p for some prime p dividing gcd(|G|, |H|). We use the well-known fact that finite nilpotent groups are direct products of their Sylow subgroups [DF04, Ch 6, Theorem 3].
Theorem 4.3. If G is a finite group, H is a finite nilpotent group, and p is the smallest prime divisor of gcd(|G|, |H|) such that G has a normal subgroup of index p, then Proof. The lower bound follows from Lemma 3.8 so it suffices to show the upper bound. Write H = P 1 × · · · × P r where P i is the Sylow p i -subgroup of H, and the p i are distinct. If p i < p, then G has no normal subgroup of index p i by assumption, so by Proposition 4.2 it follows that Λ G,P i = 0. On the other hand, if G has a normal subgroup of index p i , then it follows from Proposition 2.6 and Lemma 3.8 that Λ G,P i = 1 p i . Therefore, by Proposition 3.6, it follows that Λ G,H = max i Λ G,P i = 1 p .

Solvable domain
In this section, we prove Theorem 1.1 when G is solvable. As in Section 4, we begin by considering the case where G has no normal subgroups of index p for any prime p dividing gcd(|G|, |H|). which divides |H|, so p divides gcd(|G|, |H|). By our hypothesis, N cannot exist, so φ must be trivial.
We proceed to the case where G has a normal subgroup of index p for some prime p dividing gcd(|G|, |H|). Let p be the minimal such prime, so that we wish to show Λ G,H = 1 p . We first consider the special case where every prime divisor of |G| less than p also divides |H|. In this case, we show that G has no subgroups of index less than p, which yields the upper bound. To show this, we use the following fact, due to Berkovich, found as an exercise in [Isa08].
Proposition 5.2 ([Isa08, Exercise 3B.15]). Let G be a finite solvable group. Suppose H < G is a proper subgroup of G with smallest index. Then H ⊳ G.
We now prove the upper bound for the special case.
Lemma 5.3. Suppose G is a finite solvable group, H is any group, and p is the smallest prime divisor of gcd(|G|, |H|) such that G has a normal subgroup of index p. If every prime less than p dividing |G| also divides |H|, then Proof. We claim that G has no subgroups of index less than p. Let S be the subgroup with smallest possible index. By Proposition 5.2, S is normal. Since S is a maximal normal subgroup, by Proposition 4.1 it follows that the index [G : S] = q for some prime q dividing |G|. If q < p, then our hypotheses imply that q divides |H|, so G has a normal subgroup of prime index less than p dividing |H|, contradicting the minimality of p. Thus [G : S] ≥ p, proving our claim. By Lemma 3.8, | Hom(G, H)| > 1, so by Proposition 2.5, there exist homomorphisms φ, ψ ∈ Hom(G, H) such that agr(φ, ψ) = Λ G,H . By Proposition 2.1, Eq(φ, ψ) is a subgroup of G, so it follows that Λ G,H = agr(φ, ψ) = 1/[G : Eq(φ, ψ)] ≤ 1/p.
We deal with the general case using the following theorem of Hall [Hal38] characterizing finite solvable groups as those with Sylow bases.
Theorem 5.4 ( [Hal38]). Let G be a finite group with order prime factorization |G| = m i=1 p e i i . Then G is solvable if and only if it has Sylow p i -subgroups P i such that G = P 1 ⊲⊳ · · · ⊲⊳ P m .
We use this decomposition to filter out all the prime divisors of |G| not dividing |H| to reduce to our special case.
Theorem 5.5. If G is a finite solvable group, H is any group, and p is the smallest prime divisor of gcd(|G|, |H|) such that G has a normal subgroup of index p, then Proof. The lower bound follows from Lemma 3.8 so it suffices to show the upper bound. By Hall's theorem (Theorem 5.4), we can write G = G 1 ⊲⊳ G 2 where gcd(|G 2 |, |H|) = 1 and every prime dividing |G 1 | divides |H|. By Proposition 3.9, | Hom(G 2 , H)| = 1. Let N ⊳ G be the smallest normal subgroup of G containing G 2 . By Proposition 3.3, Λ G,H = Λ G/N,H , so it suffices to upper bound Λ G/N,H .
Since |G 2 | divides |N |, it holds that [G : N ] divides [G : G 2 ] = |G 1 |. In particular, every prime dividing |G/N | divides |H|. Moreover, G/N has no normal subgroups of index q < p, for if it did, it would follow from the Lattice Theorem that G has a normal subgroup of index q, and moreover q divides gcd(|G|, |H|), contradicting the minimality of p. Thus, G/N has no normal subgroups of index less than p. Thus, by Lemma 5.3, it follows that Λ G/N,H ≤ 1 p .
The formula for Λ G,H for solvable G does not extend to arbitrary finite groups for the obvious reason that G may not have any normal subgroups of prime index. This holds, for instance, if G is any non-abelian simple group. One might then hope that the modified statement, where we drop the requirement that p be prime, holds. For simple G, this formula would be Λ G,H = 1 |G| . However, the following is a simple (pun intended) counterexample.

Non-abelian simple groups
We would like to determine Λ G,H for arbitrary finite groups G and H. We propose a two-part strategy for doing this. First, understand Λ G,H for simple groups G. Then, understand how to determine Λ G,H for arbitrary G by cleverly decomposing G. In Section 3, we proved some general facts about Λ G,H which could be useful (but far from complete) for the second part of this program. In this section, we explore the first part, namely we investigate Λ G,H for non-abelian simple groups G. A full investigation would entail using the classification of finite simple groups and considering each family of finite simple groups, which we do not do in this work. Instead, we prove some nontrivial lower bounds on Λ G,H for general non-abelian simple G. We then prove some lower and upper bounds on Λ G,G for the specific family {A n } n≥5 of alternating groups and pin down Λ A 5 ,A 5 = 1 10 exactly. We highlight a major difficulty, which is that in the general setting, unlike in the setting where G is solvable, Λ G,H depends on how copies of G are embedded in H, not just on the prime divisors of |G| and |H| and the normal subgroup structure of G.

Domain and codomain are isomorphic
If H does not contain a subgroup isomorphic to G, then Hom(G, H) is trivial. Let us assume that |G| . Can we achieve better agreement? Better agreement must come from two automorphisms φ, ψ ∈ Aut(G). Note that φ(g) = ψ(g) if and only if (φ −1 • ψ) ∈ Aut(G) fixes g, so we wish to find a non-identity automorphism φ ∈ Aut(G) is the subset of G fixed by φ ∈ Aut(G). Observe that the group Aut(G) naturally acts on the set G via φ · g = φ(g). Let G/ Aut(G) denote the orbits of G under this group action. By Burnside's lemma, Since G id = G, where id ∈ Aut(G) is the identity automorphism, By averaging, this implies that there is some non-identity automorphism φ ∈ Aut(G) such that and thus, by dividing by |G|, we have

Alternating groups
In this section, we prove the following.
When n = 6, the upper bound is strict.
For n = 5, the lower bound is tight, that is Λ A 5 ,A 5 = |S 3 | |A 5 | = 1 10 . This is because the only subgroups of A n larger than S 3 , up to isomorphism, are the dihedral group D 10 of order 10 generated by (1 2 3 4 5) and (2 5)(3 4), and A 4 . One can check that no conjugation fixes all of A 4 nor all of D 10 .
For the proof of Proposition 6.1, we use the following fact.
Claim 6.2. Let n ≥ 3. The subgroup A n−1 ≤ A n is the unique subgroup (up to isomorphism) of A n of smallest index. That is, there are no subgroups of A n with index less than n, and any subgroup of index n is isomorphic to A n−1 .
Proof. First, we show that there are no subgroups of index less than n. Suppose H ≤ A n with m [A n : H] < n. The group A n acts on the left cosets A n /H by left multiplication, i.e. there is a homomorphism ρ : A n → Perm(A n /H) ∼ = S m . This action is clearly nontrivial, and since A n is simple, this means ρ is injective, so A n embeds into S m . This is impossible since n > 2 implies |A n | = n! 2 > (n − 1)! ≥ m! = |S m |. Now, we show uniqueness up to isomorphism. Let H ≤ A n have index n. We will show that H ∼ = A n−1 . Again, consider the action ρ as defined above. We established that A n acts faithfully on A n /H. Observe that H acts on A n /H by fixing the coset H and permuting the other n − 1 cosets. Therefore, ρ(H) is a subgroup of a copy of A n−1 inside Perm(A n /H). Since ρ is injective, |ρ(H)| = (n − 1)!, and so ρ(H) is actually isomorphic to A n−1 . Moreover, H is isomorphic to ρ(H) by the injectivity of ρ, so H is isomorphic to A n−1 .
Proof of Proposition 6.1. For the lower bound, note that there is a twisted copy of S n−2 inside A n , generated by the elements (1 2 · · · n − 2) and (n − 1 n) when n is even, and by (1 2 · · · n − 1) and (n − 1 n) when n is odd. In either case, the automorphism φ ρ : σ → ρσρ −1 with ρ = (n − 1 n) fixes this copy of S n−2 .
The upper bound follows from the fact that A n−1 is the unique subgroup (up to isomorphism) of A n of smallest index (Claim 6.2). For n = 6, every automorphism of A n is conjugation by some σ ∈ S n , but no σ ∈ S n fixes every element of A n−1 .