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Note Mat. 39 (2019) no. 2, 87–93. doi:10.1285/i15900932v39n2p87

Normal subgroups of finite non-abelian metacyclic p-groups of class two of odd order

Pradeep Kumar

Department of Mathematics, Central University of South Bihar, India 14p.shaoran@gmail.com

Received: 23.2.2019; accepted: 8.10.2019.

Abstract. In this paper, we determine the normal subgroups of a finite non-abelian meta- cyclic p-group of class two for odd prime p.

Keywords:Metacyclic p-groups, Subgroups, Normal subgroups

MSC 2000 classification:primary 20D15, secondary 20D60

Introduction

Determining the subgroups of finite groups is one of the important problems in finite group theory. In last century, the problem was completely solved for finite abelian groups (see [2, 5]). In [3], Calhoun determined the subgroups of ZM-groups (finite group with all Sylow subgroups are cyclic). Motivated by the paper of Calhoun [3], M. T˘arn˘auceanu determined the normal subgroups of ZM-groups [9]. In [9], T˘arn˘auceanu also proposed the following problem:

“Describe the normal subgroups of an arbitrary metacyclic group”.

In this paper, we partially answer the above problem by determining the normal subgroups of finite non-abelian metacyclic p-groups of class two (p odd).

A group G is said to be metacyclic if it contains a normal cyclic subgroup C with cyclic quotient group G/C. Throughout the paper groups will always be finite and N denotes the set of positive integers.

1 Basic Results

In this section, we give some results that will be needed later. First, we state a theorem that gives a presentation for a non-abelian metacyclic p-group of class two, p odd.

http://siba-ese.unisalento.it/ c 2019 Universit`a del Salento

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Theorem 1 ([1, Theorem 1.1]). Let M be a non-abelian metacyclic p-group of class two, p > 2. Then M is isomorphic to the following group:

M ∼= hx, y | xpr = yps = 1, [x, y] = xpr−δi, where r, s, δ ∈ N = { 1, 2, . . . }, r ≥ 2δ, s ≥ δ ≥ 1.

Lemma 1 ([8]). Let G be a p-group of class two, and let x, y, z ∈ G. Then (i) [xy, z] = [x, z][y, z],

(ii) [xm, y] = [x, ym] = [x, y]m, for any integer m.

Now, we state the Goursat’s Lemma related to the subgroups of the direct product of two groups.

Proposition 1 ([4, Goursat’s Lemma]). Let X and Y be arbitrary groups.

Then there is a bijection between the set of all subgroups of X × Y and the set T of all 5-tuples (A, A0, B, B0, φ), where A0 E A ≤ X, B0 E B ≤ Y and φ : A/A0 −→ B/B0 is an isomorphism. More precisely, the subgroup corresponding to (A, A0, B, B0, φ) is

H = { (x, y) ∈ A × B | φ(xA0) = yB0}. (1.1) 1.1 Subgroups of Zpr × Zps

In this subsection, we give a representation of subgroups of Zpr× Zps given in [10]. With out loss of generality, let r ≥ s ≥ 1. With the notations used in Proposition 1, let X = Zpr = hxi and Y = Zps = hyi. Let |A| = pu, |A0| = pv, |B| = pq, |B0| = pt.

Further, let A ≤ X, A = hxpr−ui, where 0 ≤ u ≤ r, A0 ≤ A, and A0 = hxpr−vi, where 0 ≤ v ≤ u. Then A/A0 = hxpr−uA0i. Similarly, let B ≤ Y, B = hyps−qi, where 0 ≤ q ≤ s, B0 ≤ B, and B0 = hyps−ti, where 0 ≤ t ≤ q. Then B/B0= hyps−qB0i.

Again, for |A/A0| = |B/B0| that is, u − v = q − t, the isomorphisms φl : A/A0 → B/B0 are given by

φl(xipr−uA0) = yilps−qB0, where 1 ≤ l ≤ pu−v with gcd(l, p) = 1.

Using Proposition 1, one can deduce that the subgroups H of Zpr× Zps are of the form H = { xipr−uyjps−q ∈ A×B | (il −j)ps−q ≡ 0 mod ps−t} (for details see [7]).

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Now, let

Spr,ps :={ (u, v, q, t, l) | 0 ≤ u ≤ r, 0 ≤ v ≤ u, 0 ≤ q ≤ s, 0 ≤ t ≤ q, u − v = q − t, 1 ≤ l ≤ pu−v, and gcd(l, p) = 1 }.

For (u, v, q, t, l) ∈ Spr,ps, define

Hu,v,q,t,l={ xipr−uyjps−q | (il − j)ps−q ≡ 0 mod ps−t, 1 ≤ i ≤ pu, and

1 ≤ j ≤ pq}. (1.2)

Theorem 2 ([10, Theorem 3.1]). The map (u, v, q, t, l) 7→ Hu,v,q,t,l is a bijection between the set Spr,ps and the set of subgroups of Zpr × Zps, where r, s ∈ N.

2 Representation of Normal Subgroups of Finite Non- Abelian Metacyclic p-Groups

In this section, first we determine the subgroups of non-abelian metacyclic p-groups M of class two (p odd). For this, we use Baer’s trick to construct an abelian group Mw corresponding to M . Then we show that there is a one-one correspondence between subgroups of M and Mw.

Let G be a group. If we can define a binary operation ◦ on G by x ◦ y = w(x, y)

where w is some fixed word in x, y ∈ G such that the set G forms a group with operation ◦, then we say w to be a group-word for G, and we write the corresponding group by Gw, that is, as a set Gw = G and operation of Gw is ◦.

Now if G is a p-group of class two, p odd, then we can define a group- word w for G as follows; for x, y ∈ G, w(x, y) = x ◦ y := xy[x, y]m−12 (where [x, y] = x−1y−1xy and m is the exponent of γ2(G), the commutator subgroup of group G). Moreover, x ◦ y = xy[x, y]m−12 = yx[x, y]m+12 = yx[y, x]m−12 = y ◦ x.

Thus the corresponding group Gw is abelian (for more details see [6, p. 142] and [7]). Now onward by w, we mean the group word defined as above, M denotes a non-abelian metacyclic p-group of class two (p odd), n = m−12 where m is the exponent of γ2(M ), and Mw is the corresponding abelian group of M defined as above.

Proposition 2. The corresponding abelian group of M is given by Mw ∼= hx, y | xpr = yps = 1, xy = yxi.

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Proof. Let K = hx, y | xpr = yps = 1, xy = yxi. As a set Mw = M . Now, take an element g = xiyj ∈ M . So g = xi ◦ yj ◦ [xi, yj]−n = xi−pr−δijn◦ yj. Therefore, Mw = hx, yi. Since powers of each element in M and Mware same, so xpr = yps = 1. Also, x◦y = y◦x. Thus, the generators of Mw satisfy the relations of K, so by Von Dyck’s Theorem [8, p. 51], there is a surjective homomorphism φ : K −→ Mw with x → x and y → y. Moreover, |Mw| = |M | = pr+s. So,

|Mw| = |K|. Thus, Mw ∼= K. This completes the proof. QED Note that to avoid ambiguity of operations, we write

Mw = hx, y | xpr = yps = 1, x ◦ y = y ◦ xi.

It is clear that Mw∼= Zpr × Zps.

Lemma 2. A subset of M is a subgroup of M if and only if it is a subgroup of Mw.

Proof. It is not hard to see that subgroups of M are subgroups of Mw. For converse, consider an arbitrary subgroup H of Mw. Using equation (1.2), the subgroup H is of the form

H = { xipr−u◦ yjps−q | il ≡ j mod pq−t, 1 ≤ i ≤ pu, and 1 ≤ j ≤ pq}, where q > t and for q = t,

H = { xipr−u◦ yjps−q | 1 ≤ i ≤ pu and 1 ≤ j ≤ pq}.

Now, take g1, g2 ∈ H, where g1 = xipr−u◦ yjps−q, and g2 = xi0pr−u◦ yj0ps−q. To show that H is also a subgroup of M , it is sufficient to show that H is closed with the operation of M , that is, g1g2 ∈ H.

We have g1g2 = g1 ◦ g2 ◦ [g1, g2]−n, where n = m−12 , m is the exponent of γ2(M ). Further, [g1, g2] = [xipr−u◦ yjps−q, xi0pr−u◦ yj0ps−q], that is, in turn, equivalent to

[g1, g2] = [x, y](ij0−ji0)pr+s−u−q (Lemma 1).

Now,

g1g2= xipr−u◦ yjps−q ◦ xi0pr−u◦ yj0ps−q ◦ [x, y]−n(ij0−ji0)pr+s−u−q

= x{ i+i0−n(ij0−ji0)pr−δ+s−q}pr−u◦ y{ j+j0}ps−q ([x, y] = xpr−δ).

For q = t, it is evident that g1g2 ∈ H. Now, assume that q > t. Since g1, g2 ∈ H, the following equations hold

il ≡ j mod pq−t, (2.1)

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i0l ≡ j0 mod pq−t. (2.2) Using (2.1) & (2.2), we deduce that il + i0l ≡ j + j0 mod pq−tand (ij0− ji0)l ≡ 0 mod pq−t. Since gcd(l, p) = 1, we get (ij0−ji0) ≡ 0 mod pq−t. Thus, we conclude that

{ i + i0− n(ij0− ji0)pr−δ+s−q}l ≡ j + j0 mod pq−t.

Thus, g1g2 ∈ H. This completes the proof. QED

Now, we determine the normal subgroups of non-abelian metacyclic group M of class two (p odd).

By Theorem 1, we have

M ∼= hx, y | xpr = yps = 1, [x, y] = xpr−δi, (∗)

where r, s, δ ∈ N, r ≥ 2δ, s ≥ δ ≥ 1. By subsection 1.1 and Lemma 2, the subgroups of M are of the form

Hu,v,q,t,l∼={ xipr−u◦ yjps−q | (il − j)ps−q ≡ 0 mod ps−t, 1 ≤ i ≤ pu, and 1 ≤ j ≤ pq},

where, (u, v, q, t, l) ∈ Spr,ps.

Lemma 3. For (u, v, q, t, l) ∈ Spr,ps, the subgroup Hu,v,q,t,lof M is a normal subgroup if and only if u − δ + s − q ≥ q − t and r − δ ≥ q − t.

Proof. The subgroup Hu,v,q,t,l is a normal subgroup if and only if g−1hg ∈ Hu,v,q,t,l for every g ∈ M and h ∈ Hu,v,q,t,l. Take an element g = xayb ∈ M and h = xipr−u◦ yjps−q ∈ Hu,v,q,t,l. Now, we have g−1hg = h[h, g] = h ◦ [h, g]. Thus

g−1hg = h ◦ [h, g]

= xipr−u◦ yjps−q◦ [xipr−u◦ yjps−q, xayb]

= xipr−u◦ yjps−q◦ [x, y]ibpr−u−ajps−q (Lemma 1)

= xipr−u+pr−δ(ibpr−u−ajps−q)◦ yjps−q ([x, y] = xpr−δ).

Let Hu,v,q,t,l be a normal subgroup of M . Now, if g−1hg ∈ Hu,v,q,t,l, then ipr−u+ pr−δ(ibpr−u− ajps−q) ≡ 0 mod pr−u,

and that is equivalent to ajpr−δ+s−q ≡ 0 mod pr−u. The latter equation must hold for every possible a, j. Thus pr−u | pr−δ+s−q. So, r − δ + s − q ≥ r − u.

This implies u − δ + s − q ≥ 0. Now, assume that u − δ + s − q ≥ 0, then g−1hg = x(i+ibpr−δ−ajpu−δ+s−q)pr−u◦yjps−q. Now, if g−1hg ∈ H, then [(i+ibpr−δ

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ajpu−δ+s−q)l − j]ps−q ≡ 0 mod ps−t. For q = t, the latter equation always holds. Now, suppose q > t, then (i + ibpr−δ− ajpu−δ+s−q)l ≡ j mod pq−t. Since h ∈ H, so il ≡ j mod pq−t. Thus we have that (ibpr−δ − ajpu−δ+s−q)l ≡ 0 mod pq−t. Since gcd(l, p) = 1, (ibpr−δ− ajpu−δ+s−q) ≡ 0 mod pq−t. The latter equation must hold for every a, b and i, j such that h ∈ H, g ∈ M . This implies pu−δ+s−q ≡ 0 mod pq−t and pr−δ ≡ 0 mod pq−t. Thus u − δ + s − q ≥ q − t and r − δ ≥ q − t. It is not hard to see that converse part holds. This completes

the proof. QED

Now, for every r, s, δ ∈ N such that r ≥ 2δ, s ≥ δ ≥ 1, let

Jp0r,ps :={ (u, v, q, t, l) | 0 ≤ u ≤ r, 0 ≤ v ≤ u, 0 ≤ q ≤ s, 0 ≤ t ≤ q, u − v = q − t, 1 ≤ l ≤ pu−v, gcd(l, p) = 1, u − δ + s − q ≥ q − t, and r − δ ≥ q − t }.

For (u, v, q, t, l) ∈ Jp0r,ps, define

Nu,v,q,t,l:={ xipr−uyjps−q[x, y]nijpr−u+s−q | (il − j)ps−q ≡ 0 mod ps−t, 1 ≤ i ≤ pu, and 1 ≤ j ≤ pq}.

Theorem 3. The map (u, v, q, t, l) 7→ Nu,v,q,t,l is a bijection between the set Jp0r,ps and the set of normal subgroups of non-abelian metacyclic p-group M of class two (p odd) as in (∗).

Proof. This follows from Lemmas 2, 3 and Theorem 2. QED Acknowledgements. The author would like to thank the referee for his very valuable comments and suggestions.

References

[1] J. R. Beuerle: An elementary classification of finite metacyclic p-groups of class at least three, Algebr. Colloq. 12 (2005), 553–562.

[2] G. Bhowmik: Evaluation of divisor functions of matrices, Acta Arith. 74 (1996), 155–154.

[3] W. C. Calhoun: Counting the subgroups of some finite groups, Amer. Math. Monthly 94 (1987), 54–59.

[4] R. R. Crawford, K. D. Wallace: On the number of subgroups of index two-an appli- cation of Goursat’s theorem for groups, Math. Mag. 48 (1975), 172–174.

[5] S. Delsarte: Fonctions de m¨obius sur les groupes abeliens finis, Ann. of Math. 49 (1948), 600–609.

[6] I. M. Isaacs: Finite Group Theory, Graduate Studies in Mathematics, Vol. 92, Amer.

Math. Soc. Providence, 2008.

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[7] P. Kumar, V. K. Jain: On the number of subgroups of non-metacyclic minimal non- abelian p-groups, Asian-Eur. J. Math, doi: 10.1142/S1793557120500941.

[8] D. J. S. Robinson: A Course in the Theory of Groups, 2nd Edition, Springer, New York, 1996.

[9] M. T˘arn˘auceanu, The normal subgroup structure of ZM-groups, Ann. Mat. Pura Appl.

193 (2014), 1085–1088.

[10] L. T´oth, Subgroups of finite abelian groups having rank two via Goursat’s lemma, Tatra Mt. Math. Publ. 59 (2014), 93–103.

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