Bases in Dihedral and Boolean Groups

Volodymyr Gavrylkiv Vasyl Stefanyk Precarpathian National University Department of Algebra and Geometry Shevchenko str., 57 Ivano-Frankivsk Ukraine

                     vgavrylkiv@gmail.com

Abstract A subset B of a group G is called a basis of G if G = B^{2}. The smallest cardinality of a basis of G is called the basis size of G. We prove upper bounds for basis sizes of dihedral and Boolean groups. We find a lower bound for the basis size of a Boolean group. We also calculate basis sizes for dihedral and Boolean groups of small orders.

1 Introduction

A subset B of a group G is called a basis of G if each element g ∈ G can be written as g = ab for some a, b ∈ B. The smallest cardinality of a basis of G is called the basis size of G and is denoted by r[G]. The problem of estimating r[G] for a cyclic group G was first proposed by Schur and various bounds were obtained by Rohrbach [13], Moser [10], Stöhr [15], Klotz [7] and others. Bases for arbitrary groups were dealt by Rohrbach [14] and lately by Cherly [5], Bertram and Herzog [4], Nathanson [11], Kozma and Lev [8].

A family G of finite groups is well-based if there exists a constant c ∈ ℝ_{+} such that r[G] ≤ c\sqrt{|G|} for each G ∈ G. Bertram and Herzog [4] showed that the families of nilpotent groups, as well as the families of the alternating and symmetric groups, are well-based. Kozma and Lev [8] proved (using the classification of finite simple groups) that r[G] ≤ \frac{4}{\sqrt{3}} \sqrt{|G|} for any finite group G. Therefore, the family of all finite groups is well-based.

The definition of a basis B for a group G implies that |G| ≤ |B|^{2}, and hence r[G] ≥ \sqrt{|G|}. The fraction

δ[G] := \frac{r[G]}{\sqrt{|G|}} ≥ 1

is called the basis characteristic of G.

In this paper we shall evaluate basis characteristics of dihedral and Boolean groups. We recall that the dihedral group D_{2n} of order 2n is the isometry group of a regular n-gon. The dihedral group D_{2n} contains a normal cyclic subgroup of index 2. A standard model of a cyclic group of order n is the multiplicative group

C_{n} = {z ∈ ℂ : z^{n} = 1}

of n-th roots of 1. The group C_{n} is isomorphic to the additive group of the ring ℤ_{n} = ℤ/nℤ. A group G is called Boolean if g^{−1} = g for every g ∈ G. It is well-known that Boolean groups are Abelian and are isomorphic to the powers of the two-element cyclic group C_{2}.

Kozma and Lev [8] proved that each finite group G has the basis characteristic δ[G] ≤ \frac{4}{\sqrt{3}} ≈ 2.3094. Until now, the best estimate for an upper bound of the basis characteristic was the estimate for the class of cyclic groups: each cyclic group C_{n} has the basis characteristic δ[C_{n}] ≤ 2. We shall show that for dihedral groups this upper bound can be improved to δ[D_{2n}] ≤ \frac{24}{\sqrt{146}} ≈ 1.9862. Moreover, if n ≥ 2 · 10^{15}, then δ[D_{2n}] < \frac{4}{\sqrt{6}} ≈ 1.633. Also we prove

that each Boolean group G has the basis size ^{\frac{1+\sqrt{8|G|−7}}{2}} ≤ r[G] < \frac{3}{\sqrt{2}} \sqrt{|G|} and, therefore, its basis characteristic δ[G] < \frac{3}{\sqrt{2}} ≈ 2.1213.

For a class G of finite groups the number

δ[G] = sup_{G∈G} δ[G]

is called the basis characteristic of the class G.

For a real number x we put

⌈x⌉ = min{n ∈ ℤ : n ≥ x} and ⌊x⌋ = max{n ∈ ℤ : n ≤ x}.

2 Known results

In this section we recall some known results on bases in finite groups. Kozma and Lev [8] proved the following fundamental fact.

Theorem 1. Each finite group G has the basis characteristic δ[G] ≤ \frac{4}{\sqrt{3}}.

Bertram and Herzog [4] proved the following proposition.

Proposition 2. Let G be a finite group. Then

1) r[G] ≤ |G : H| · r[H] for any subgroup H ⊂ G;

2) r[G] ≤ r[G/H] · |H| for any normal subgroup H ⊂ G;

3) r[G] ≤ 2r[G/H] · r[H] for any normal subgroup H ⊂ G; 4) r[G] ≤ |H| + |G : H| − 1 for any subgroup H ⊂ G.

In evaluating basis characteristics of dihedral groups we shall use difference characteristics of cyclic groups. A subset B of a group G is called a difference basis of G if each element g ∈ G can be written as g = xy^{−1} for some x, y ∈ B. The smallest cardinality of a difference basis of G is called the difference size of G and is denoted by Δ[G]. The definition of a difference basis B for a group G implies that |G| ≤ |B|^{2}, and hence Δ[G] ≥ \sqrt{|G|}. The fraction

ð[G] := \frac{Δ[G]}{\sqrt{|G|}} ≥ 1

is called the difference characteristic of G.

Difference basis have applications in the study of structure of superextensions of groups, see [1, 3].

Difference sizes of finite cyclic groups were evaluated in [2] with the help of difference sizes of the order-intervals [1, n]∩ℤ in the additive group ℤ of integer numbers. For a natural number n ∈ ℕ by Δ[n] we shall denote the difference size of the order-interval [1, n] ∩ ℤ and by ð[n] := \frac{Δ[n]}{\sqrt{n}} its difference characteristic. The asymptotics of the sequence (ð[n])_{n=1}^{∞} was studied by Rédei and Rényi [12], Leech [9] and Golay [6] who eventually proved that

\sqrt{2 + \frac{4}{3π}} < \sqrt{2 + max_{0<φ<2π} \frac{2 sin(φ)}{φ+π}} ≤ lim_{n→∞} ð[n] = inf_{n∈ℕ} ð[n] ≤ ð[6166] = \frac{128}{\sqrt{6166}} < ð[6] = \sqrt{\frac{8}{3}}.

Banakh and Gavrylkiv [2] applied difference sizes of the order-intervals [1, n] ∩ ℤ to give upper bounds for difference sizes of finite cyclic groups.

Proposition 3. For every n ∈ ℕ the cyclic group C_{n} has the difference size Δ[C_{n}] ≤ Δ[⌈\frac{n−1}{2}⌉], which implies that

lim sup_{n→∞} ð[C_{n}] ≤ \frac{1}{\sqrt{2}} inf_{n∈ℕ} ð[n] ≤ \frac{64}{\sqrt{3083}} < \frac{2}{\sqrt{3}}.

Banakh and Gavrylkiv [2] proved the following upper bound for difference characteristics of cyclic groups.

Theorem 4. For any n ∈ ℕ the cyclic group C_{n} has the difference characteristic:

1) ð[C_{n}] ≤ ð[C_{4}] = \frac{3}{2};

2) ð[C_{n}] ≤ ð[C_{2}] = ð[C_{8}] = \sqrt{2} if n ≠ 4;

3) ð[C_{n}] ≤ \frac{12}{\sqrt{73}} < \sqrt{2} if n ≥ 9;

4) ð[C_{n}] ≤ \frac{24}{\sqrt{293}} < \frac{12}{\sqrt{73}} if n ≥ 9 and n ≠ 292;

5) ð[C_{n}] < \frac{2}{\sqrt{3}} if n ≥ 2 · 10^{15}.

The following Table 1 of difference sizes and characteristics of cyclic groups C_{n} for ≤ 100 is taken from [2].

\begin{matrix}n & Δ[C_{n}] & ð[C_{n}] & n & Δ[C_{n}] & ð[C_{n}] & n & Δ[C_{n}] & ð[C_{n}] & n & Δ[C_{n}] & ð[C_{n}] \\ 1 & 1 & 1 & 26 & 6 & 1.1766… & 51 & 8 & 1.1202… & 76 & 10 & 1.1470… \\ 2 & 2 & 1.4142… & 27 & 6 & 1.1547… & 52 & 9 & 1.2480… & 77 & 10 & 1.1396… \\ 3 & 2 & 1.1547… & 28 & 6 & 1.1338… & 53 & 9 & 1.2362… & 78 & 10 & 1.1322… \\ 4 & 3 & 1.5 & 29 & 7 & 1.2998… & 54 & 9 & 1.2247… & 79 & 10 & 1.1250… \\ 5 & 3 & 1.3416… & 30 & 7 & 1.2780… & 55 & 9 & 1.2135… & 80 & 11 & 1.2298… \\ 6 & 3 & 1.2247… & 31 & 6 & 1.0776… & 56 & 9 & 1.2026… & 81 & 11 & 1.2222… \\ 7 & 3 & 1.1338… & 32 & 7 & 1.2374… & 57 & 8 & 1.0596… & 82 & 11 & 1.2147… \\ 8 & 4 & 1.4142… & 33 & 7 & 1.2185… & 58 & 9 & 1.1817… & 83 & 11 & 1.2074… \\ 9 & 4 & 1.3333… & 34 & 7 & 1.2004… & 59 & 9 & 1.1717… & 84 & 11 & 1.2001… \\ 10 & 4 & 1.2649… & 35 & 7 & 1.1832… & 60 & 9 & 1.1618… & 85 & 11 & 1.1931… \\ 11 & 4 & 1.2060… & 36 & 7 & 1.1666… & 61 & 9 & 1.1523… & 86 & 11 & 1.1861… \\ 12 & 4 & 1.1547… & 37 & 7 & 1.1507… & 62 & 9 & 1.1430… & 87 & 11 & 1.1793… \\ 13 & 4 & 1.1094… & 38 & 8 & 1.2977… & 63 & 9 & 1.1338… & 88 & 11 & 1.1726… \\ 14 & 5 & 1.3363… & 39 & 7 & 1.1208… & 64 & 9 & 1.125 & 89 & 11 & 1.1659… \\ 15 & 5 & 1.2909… & 40 & 8 & 1.2649… & 65 & 9 & 1.1163… & 90 & 11 & 1.1595… \\ 16 & 5 & 1.25 & 41 & 8 & 1.2493… & 66 & 10 & 1.2309… & 91 & 10 & 1.0482… \\ 17 & 5 & 1.2126… & 42 & 8 & 1.2344… & 67 & 10 & 1.2216… & 92 & 11 & 1.1468… \\ 18 & 5 & 1.1785… & 43 & 8 & 1.2199… & 68 & 10 & 1.2126… & 93 & 12 & 1.2443… \\ 19 & 5 & 1.1470… & 44 & 8 & 1.2060… & 69 & 10 & 1.2038… & 94 & 12 & 1.2377… \\ 20 & 6 & 1.3416… & 45 & 8 & 1.1925… & 70 & 10 & 1.1952… & 95 & 12 & 1.2311… \\ 21 & 5 & 1.0910… & 46 & 8 & 1.1795… & 71 & 10 & 1.1867… & 96 & 12 & 1.2247… \\ 22 & 6 & 1.2792… & 47 & 8 & 1.1669… & 72 & 10 & 1.1785… & 97 & 12 & 1.2184… \\ 23 & 6 & 1.2510… & 48 & 8 & 1.1547… & 73 & 9 & 1.0533… & 98 & 12 & 1.2121… \\ 24 & 6 & 1.2247… & 49 & 8 & 1.1428… & 74 & 10 & 1.1624… & 99 & 12 & 1.2060… \\ 25 & 6 & 1.2 & 50 & 8 & 1.1313… & 75 & 10 & 1.1547… & 100 & 12 & 1.2 \end{matrix}

Table 1: Difference sizes and characteristics of cyclic groups C_{n} for n ≤ 100.

3 Basis sizes and characteristics of dihedral groups

In this section we shall evaluate basis sizes and characteristics of dihedral groups. The following proposition yields some upper bounds for the basis size of a dihedral group.

Proposition 5. For any numbers n, m ∈ ℕ the dihedral group D_{2nm} has the basis size

1) r[D_{2nm}] ≤ 2n · r[C_{m}];

2) r[D_{2nm}] ≤ r[D_{2n}] · m;

3) r[D_{2nm}] ≤ 2r[D_{2n}] · r[C_{m}].

Proof. It is well-known that the dihedral group D_{2nm} contains a normal cyclic subgroup of order nm, which can be identified with the cyclic group C_{nm}. The subgroup C_{m} ⊂ C_{nm} is normal in D_{2mn} and the quotient group D_{2mn}/C_{m} is isomorphic to D_{2n}. Applying Proposition 2(1-3), we obtain the desired upper bounds.

Theorem 6. For any n ∈ ℕ the dihedral group D_{2n} has the basis size r[D_{2n}] ≤ 2Δ[C_{n}].

Proof. It is well-known that the group D_{2n} contains a cyclic subgroup C_{n} and an element s ∈ D_{2n} \ C_{n} such that s = s^{−1} and sxs = x^{−1} for all x ∈ C_{n}. Fix a difference basis B ⊂ C_{n} of cardinality |B| = Δ[C_{n}]. Therefore, BB^{−1} = C_{n} and sBs = B^{−1}. We claim that B^{−1} ∪sB is a basis. Indeed,

(B^{−1} ∪ sB)(B^{−1} ∪ sB) = B^{−1}B^{−1} ∪ B^{−1}sB ∪ sBB^{−1} ∪ sBsB ⊃ s(BB^{−1}) ∪ (sBs)B
= sC_{n} ∪ B^{−1}B = (D_{2n} \ C_{n}) ∪ BB^{−1} = (D_{2n} \ C_{n}) ∪ C_{n} = D_{2n}.

Therefore, r[D_{2n}] ≤ |B^{−1} ∪ sB| = 2|B| = 2Δ[C_{n}].

In Table 2 we present the results of computer calculations of basis sizes and characteristics of dihedral groups of order ≤ 80.

\begin{matrix}2n & r[D_{2n}] & 2Δ[C_{n}] & δ[D_{2n}] & 2n & r[D_{2n}] & 2Δ[C_{n}] & δ[D_{2n}] \\ 2 & 2 & 2 & 1.4142… & 42 & 9 & 10 & 1.3887… \\ 4 & 3 & 4 & 1.5 & 44 & 9 & 12 & 1.3568… \\ 6 & 3 & 4 & 1.2247… & 46 & 9 & 12 & 1.3269… \\ 8 & 4 & 6 & 1.4142… & 48 & 9 & 12 & 1.2990… \\ 10 & 4 & 6 & 1.2649… & 50 & 9 & 12 & 1.2727… \\ 12 & 5 & 6 & 1.4433… & 52 & 10 & 12 & 1.3867… \\ 14 & 5 & 6 & 1.3363… & 54 & 10 & 12 & 1.3608… \\ 16 & 5 & 8 & 1.25 & 56 & 10 & 12 & 1.3363… \\ 18 & 6 & 8 & 1.4142… & 58 & 10 & 14 & 1.3130… \\ 20 & 6 & 8 & 1.3416… & 60 & 10 & 14 & 1.2909… \\ 22 & 7 & 8 & 1.4924… & 62 & 10 & 12 & 1.2700… \\ 24 & 6 & 8 & 1.2247… & 64 & 11 & 14 & 1.375 \\ 26 & 7 & 8 & 1.3728… & 66 & 10 & 14 & 1.2309… \\ 28 & 7 & 10 & 1.3228… & 68 & 11 & 14 & 1.3339… \\ 30 & 7 & 10 & 1.2780… & 70 & 10 & 14 & 1.1952… \\ 32 & 7 & 10 & 1.2374… & 72 & 11 & 14 & 1.2963… \\ 34 & 8 & 10 & 1.3719… & 74 & 11 & 14 & 1.2787… \\ 36 & 8 & 10 & 1.3333… & 76 & 12 & 16 & 1.3764… \\ 38 & 8 & 10 & 1.2977… & 78 & 11 & 14 & 1.2455… \\ 40 & 8 & 12 & 1.2649… & 80 & 12 & 16 & 1.3416… \end{matrix}

Table 2: Basis sizes and characteristics of dihedral groups D_{2n} for 2n ≤ 80.

Theorem 7. Let D_{2n} be a dihedral group of order 2n. Then for any number n ∈ ℕ, the basis characteristic of the dihedral group is δ[D_{2n}] ≤ \frac{24}{\sqrt{146}}. If n ≥ 2 · 10^{15}, then δ[D_{2n}] < \frac{4}{\sqrt{6}}.

Proof. It is well-known that the dihedral group D_{2n} contains a normal cyclic subgroup of order n, which can be identified with the cyclic group C_{n}. Applying Theorem 6, we obtain the upper bound

δ[D_{2n}] = \frac{r[D_{2n}]}{\sqrt{2n}} ≤ \frac{2Δ[C_{n}]}{\sqrt{2n}} = \sqrt{2} · ð[C_{n}].

If n ≥ 9, then using part 3 of Theorem 4, we get ð[C_{n}] ≤ \frac{12}{\sqrt{73}}. Therefore,

δ[D_{2n}] ≤ \sqrt{2} · \frac{12}{\sqrt{73}} = \frac{24}{\sqrt{146}}.

Analyzing the data from Table 2, one can check that δ[D_{2n}] ≤ \frac{24}{\sqrt{146}} ≈ 1.9863 for all n ≤ 8.

If n ≥ 2 · 10^{15}, then using part 5 of Theorem 4, we get ð[C_{n}] < \frac{2}{\sqrt{3}}, and hence

δ[D_{2n}] ≤ \sqrt{2} · ð[C_{n}] < \frac{4}{\sqrt{6}}.

The results of computer calculations given in Table 2 and Theorem 7 imply the following lower and upper bounds for the basis characteristic of the class Dihedral of dihedral groups.

Proposition 8. The class Dihedral of dihedral groups has the basis characteristic

1.5 = \frac{3}{\sqrt{4}} = δ[D_{4}] ≤ δ[Dihedral] ≤ \frac{24}{\sqrt{146}} ≈ 1.9863.

Proof. Theorem 7 yields the upper bound

δ[Dihedral] = sup_{n∈ℕ} δ[D_{2n}] ≤ \frac{24}{\sqrt{146}}.

On the other hand, the basis size r[D_{4}] = 3 of the dihedral group D_{4} witnesses that

1.5 = δ[D_{4}] ≤ δ[Dihedral].

Question 9. Is δ[Dihedral] = δ[D_{4}] = 1.5?

4 Basis sizes and characteristics of Boolean groups

In this section we evaluate basis sizes and characteristics of finite Boolean groups. A group G is called Boolean if g^{−1} = g for every g ∈ G. It is well-known that Boolean groups are Abelian and are isomorphic to the powers of the two-element cyclic group C_{2}. Let Boolean denote the class of finite Boolean groups and let Boolean_{0} := \{G ∈ Boolean : \sqrt{|G|} ∈ ℕ\}.

We start with the lower bound for r[G].

Theorem 10. Each Boolean group G has r[G] ≥ ^{\frac{1+\sqrt{8|G|−7}}{2}}.

Proof. Take a basis B ⊂ G of cardinality |B| = r[G] and consider the surjective map ξ : B × B → G, ξ : (x, y) ↦ xy. Observe that for the unit e of the group G the preimage ξ^{−1}(e) coincides with the diagonal {(x, y) ∈ B × B : x = y} of the square B × B, and hence has cardinality |ξ^{−1}(e)| = |B|. Observe also that for any element g ∈ G \ {e} and any (x, y) ∈ ξ^{−1}(g), we get x ≠ y and yx = xy = g, which implies that |ξ^{−1}(g)| ≥ 2. Then

|B|^{2} = |B × B| = |ξ^{−1}(e)| + ∑_{g∈G\{e}} |ξ^{−1}(g)| ≥ |B| + 2(|G| − 1).

Therefore, |B|^{2}−|B|−2(|G|−1) ≥ 0, and hence r[G] = |B| ≥ ^{\frac{1+\sqrt{1+8(|G|−1)}}{2}} = ^{\frac{1+\sqrt{8|G|−7}}{2}}.

Theorem 11. Each finite Boolean group G has

\frac{1 + \sqrt{8|G| − 7}}{2} ≤ r[G] < \begin{cases}2\sqrt{|G|}, & if G ∈ Boolean_{0}; \\ \frac{3}{\sqrt{2}} \sqrt{|G|}, & otherwise. \end{cases}

Proof. The lower bound ^{\frac{1+\sqrt{8|G|−7}}{2}} ≤ r[G] follows from Theorem 10.

The group G, being Boolean, is isomorphic to (C_{2})^{n} for some n ≥ 0. Let k = ⌊\frac{n}{2}⌋ and find a subgroup H ⊂ G of order 2^{k}. By Proposition 2(4), r[G] ≤ |H| + |G/H| − 1.

If G ∈ Boolean_{0}, then n = 2k is even and

r[G] ≤ |H| + |G/H| − 1 = 2^{k} + 2^{k} − 1 < 2\sqrt{2^{2k}} = 2\sqrt{|G|}.

If G ∉ Boolean_{0}, then n = 2k + 1 is odd and

r[G] ≤ |H| + |G/H| − 1 = 2^{k} + 2^{k+1} − 1 < \frac{3}{\sqrt{2}}\sqrt{2^{2k+1}} = \frac{3}{\sqrt{2}} \sqrt{|G|}.

In the following Table 3 we present the results of computer calculations of basis sizes and characteristics of Boolean groups (C_{2})^{n} for n ≤ 6. In this table

lb[G] := ⌈\frac{1 + \sqrt{8|G| − 7}}{2}⌉ and ub[G] := ⌊\sqrt{\frac{9|G|}{2}}⌋

are the lower and upper bounds given in Theorems 10 and 11.

Theorem 11 implies the following corollaries.

\begin{matrix}G & C_{2} & (C_{2})^{2} & (C_{2})^{3} & (C_{2})^{4} & (C_{2})^{5} & (C_{2})^{6} \\ lb[G] & 2 & 3 & 5 & 6 & 9 & 12 \\ r[G] & 2 & 3 & 5 & 6 & 10 & 14 \\ ub[G] & 3 & 4 & 6 & 8 & 12 & 16 \\ δ[G] & 1.4142… & 1.5 & 1.7677… & 1.5 & 1.7677… & 1.75 \end{matrix}

Table 3: Basis sizes and characteristics of Boolean groups (C_{2})^{n} for n ≤ 6.

Corollary 12. The class Boolean of finite Boolean groups has the basis characteristic

1.7677... = \frac{5}{\sqrt{8}} = δ[(C_{2})^{3}] ≤ δ[Boolean] ≤ \frac{3}{\sqrt{2}} = 2.1213...

Proof. The upper bound δ[Boolean] ≤ \frac{3}{\sqrt{2}} follows from Theorem 11 and the lower bound δ[Boolean] ≥ δ[(C_{2})^{3}] = \frac{5}{\sqrt{8}} follows from the known value r[(C_{2})^{3}] = 5.

Corollary 13. The class Boolean_{0} has the basis characteristic

1.75 = \frac{14}{\sqrt{64}} = δ[(C_{2})^{6}] ≤ δ[Boolean_{0}] ≤ 2

Proof. The upper bound δ[Boolean_{0}] ≤ 2 follows from Theorem 11 and the lower bound δ[Boolean_{0}] ≥ δ[(C_{2})^{6}] = 1.75 follows from the known value r[(C_{2})^{6}] = 14.

References

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2010 Mathematics Subject Classification: Primary 05B10; Secondary 05E15, 20D60. Keywords: dihedral group, Boolean group, basis, basis size, basis characteristic.

Received April 18 2017; revised versions received April 20 2017; June 24 2017; June 26 2017; July 6 2017. Published in Journal of Integer Sequences, July 31 2017.

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