Ukrainian Mathematical Journal, Vol. 64, No. 10, March, 2013 (Ukrainian Original Vol. 64, No. 10, October, 2012)

ON SPLIT METACYCLIC GROUPS Yan Yang and He-guo Liu

UDC 512.5

We consider sufficient conditions for metacyclic groups to split. Specifically, we show that a finite metacyclic group G of odd order is split on its cyclic normal subgroup K if K is such that G=K is cyclic and jKj D exp G:

1. Introduction G is a metacyclic group if and only if there exists a cyclic normal subgroup K of G such that G=K is cyclic, and G D SK is called a metacyclic factorization if S is cyclic. In particular, if G has a split metacyclic factorization G D SK such that S \ K D 1; then G is called a split metacyclic group, otherwise it is called nonsplit. Hempel and Hyo-Seob Sim gave classifications of metacyclic groups in [1, 2]. Recently, the structure of the automorphism groups of metacyclic groups has been given much attention. The automorphism groups of split metacyclic p-groups were given in [3] for odd p and in [4] for p D 2: The automorphism groups of finite split metacyclic groups were given in [5]. However, the case of nonsplit groups is much more complicated than the case of split groups, and only the automorphism groups of nonsplit metacyclic p-groups of odd order were given in [6]. Thus, it is necessary to determine whether a metacyclic group is split or not before we study its automorphism. In the present paper, of particular interest are some sufficient conditions showing that finite metacyclic groups of a special type are split. 2. Notation and Preliminaries In this section, we present some general facts that will be useful in what follows. We first introduce some notation, which will be kept throughout the paper: .G/ is the set of all prime divisors of the order of a finite group G; and r.p/ is the largest integer i such that p i divides a positive integer r: G is a metacyclic group if and only if there exists a cyclic normal subgroup K of G such that G=K is cyclic. Such a subgroup K is called a kernel of G: Lemma 1. A p-group P of odd order is metacyclic with a kernel of order p and of index p ˛ if and only if it has the representation ˛

ˇ

P D ha; b j ap D b p ; b p D 1; b a D b p

ı C1

i;

where ˛; ˇ; ; and ı are positive integers such that  min.˛ C ı; ˇ C ı/: Lemma 2 ([1], Lemma 2.1). A group G is metacyclic with a kernel of order and of index ˛ if and only if it has the representation Hubei University, Wuhan, China; Hubei University of Arts and Sciences, Xiangyang, China. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 10, pp. 1432–1437, October, 2012. Original article submitted April 3, 2012. 0041-5995/13/6410–1627

c 2013

Springer Science+Business Media New York

1627

YAN YANG AND H E - GUO L IU

1628

G D ha; b j a˛ D b ˇ ; b D 1; b a D b ı i; where ˛; ˇ; ; and ı are positive integers such that jı ˛

1 and jˇ.ı

1/:

Lemma 3 ([2], Lemma 3.4). Let P D SK be a metacyclic factorization of a p-group P of odd order. The p-group P admits the representation ˛

ˇ

P D hx; y j x p D y p ; y p D 1; y x D y p

ı C1

i;

where p ˛ D jSW S \ Kj;

p ˇ D jKW S \ Kj;

p D jKj;

p ı D jKW P 0 j:

Lemma 4 ([5], Corollary 4.1). Let p be an odd prime and let k; s; and t be nonnegative integers. Then the following assertions are true: (i) .1 C p k /p

st

 1 .mod p kCs /I

(ii) if k > 0; then 1 C r s C : : : C r s.p

t

1/

 pt

.mod p t Ck /;

for r D 1 C p k : Lemma 5 ([2], Lemma 5.3). Let G be a metacyclic group with metacyclic factorization G D SK: For each set  of prime numbers, the subgroup H D S K is the unique Hall -subgroup of G such that S D S \ H and K D K \ H; and so H D .S \ H /.K \ H /: Definition 1 ([2], Definition 5.4). Let G be a group with metacyclic factorization G D SK and let  denote the set fp 2 .G/W G has a normal Hall p 0 -subgroupg: Let H denote the Hall -subgroup S K and let N denote the Hall  0 -subgroup S 0 K 0 : Then the semidirect decomposition G D HN is called the standard Hall decomposition for the metacyclic factorization G D SK: Lemma 6 ([2], Lemma 5.6). Let G D HN be a Hall decomposition for the metacyclic factorization G D SK and  D fp 2 .G/W G has a normal Hall p 0 -subgroupg: Then the following assertions are true: (i) H D S K D .S \ H /.K \ H / and H is nilpotent; (ii) N D S 0 K 0 D .S \ N /.K \ N /; K 0 D G 0 \ N; and S 0 \ K 0 D 1I (iii) S 0  CN .H /: 3. Split Metacyclic Group 3.1. Split Metacyclic p-Group Lemma 7. If P D SK is a metacyclic factorization of a p-group P of odd order, then exp P D max .jS j; jKj/:

O N S PLIT M ETACYCLIC G ROUPS

1629

Proof. By Lemma 3, P can be represented as ˛

ˇ

P D ha; bjap D b p ; b p D 1; b a D b p

ı C1

i;

where jaj D jSj and jbj D jKj: For any as b t 2 P; by virtue of Lemma 4, we have m

m

.as b t /p D asp b t.1C.p

ı C1/s C:::C.p ı C1/s.p m

1/ /

m

D asp b t .p

m Ckp ıCm /

D 1;

where p m D max .jSj; jKj/ and k is an integer. Lemma 8. P is a metacyclic p-group of odd order. If K is a normal cyclic subgroup with jKj D exp P; then P =K is cyclic. Proof. Let ˛

ˇ

P D ha; bjap D b p ; b p D 1; b a D b p

ı C1

i;

be a representation of P: Suppose that K D ham b n i and note that jKj D exp P D max.jaj; jbj/ according to Lemma 7. Then, by Lemma 4, we have .ap b p /p

1

D ap b p.1C.1Cp

ı /p C:::C.1Cp ı /p.p

1

1/ /

D 1;

whence .m; p/ D 1 or .n; p/ D 1: If .m; p/ D 1; then P =K is cyclic because P D hbiham b n i: If .n; p/ D 1; then P =K is also cyclic because P D haiham b n i: Theorem 1. Let P be a metacyclic p-group of odd order. If P has a normal cyclic subgroup K of order exp P; then P is split. Proof. P =K is cyclic by virtue of Lemma 8. Let P D HK be a metacyclic factorization of P: Then, by Lemma 3, choosing the generators x and y for H and K; respectively, we obtain the representation ˛

ˇ

P D hx; y j x p D y p ; y p D 1; y x D y p

ı C1

i;

where p ˛ D jH W H \ Kj D jP j= exp P; p D jKj D exp P;

p ˇ D jKW H \ Kj;

p ı D jKW P 0 j D jP =P 0 j=p ˛ :

If ˇ D ; then P is split. Now assume that ˇ < : Denote a D xy fp show that hai \ hyi D 1: From Lemma 1, we obtain ˛

ap D .xy fp

ˇ ˛

˛

˛

/p D x p y fp

ˇ ˛ .1C.p ı C1/C:::C.p ı C1/p ˛

1/

˛

ˇ ˛

; where f D p

D x p y fp

ˇ ˛ p˛

D yp

ˇ

ˇ Cfp ˇ

1: Then we

D yp D 1

YAN YANG AND H E - GUO L IU

1630

and ap

˛ 1

D .xy fp

ˇ ˛

/p

˛ 1

D xp

˛ 1

y fp

ˇ ˛ p˛ 1

… K;

which implies that hai \ hyi D 1: It is obvious that P D haihyi: Thus, P is split. Finally, since y a D y xy

fp ˇ ˛

ı

D y x D y 1Cp ;

we obtain the following representation of P : ˛

P D ha; y j ap D y p D 1; y a D y p

ı C1

i;

where p ˛ D jP j= exp P; p D exp P; and p ˛Cı D jP =P 0 j: Example 1. Theorem 1 may not hold for the metacyclic 2-group Q8 D hx; a j x 2 D a2 ; a4 D 1; ax D a

1

i:

Here, jaj D 4 D exp Q8 and hai C Q8 ; but Q8 is not a split group. Using Theorem 1 and Lemma 1, we can deduce the following corollary: Corollary 1. A metacyclic p-group P of odd order is nonsplit if and only if P has a representation of the form described in Lemma 1: ˛

ˇ

P D ha; b j ap D b p ; b p D 1; b a D b p

ı C1

i;

where ˛ > ˇ > ı  1 and ˇ <  ˇ C ı: 3.2. Split Metacyclic Group. In the remaining part of this section, we prove that the conclusion made above is also true for a metacyclic group of odd order. Lemma 9. Let P be a non-abelian metacyclic p-group of odd order. Suppose that both P D SK and P D S1 K1 are split metacyclic factorizations. Then the following assertions are true: (i) jS j D jS1 j and jKj D jK1 j ; (ii) P D SK1 and P D S1 K are also split metacyclic factorizations. Here, the statement that P D SK is a split metacyclic factorization means that P D SK is a metacyclic factorization and S \ K D 1: Proof. (i) Assume that ˛

P D SK D ha; b j ap D b p D 1; b a D b p p ˛1

P D S1 K1 D ha1 ; b1 j a1

p 1

D b1

ı C1

i;

p ı1 C1

D 1; b1a1 D b1

i:

O N S PLIT M ETACYCLIC G ROUPS

1631

It is obvious that jP j D p ˛C D p ˛1 C 1 and exp P D max .˛; / D max .˛1 ; 1 /: Thus, if ˛ D ˛1 ; then assertion (i) holds by virtue of Lemma 7. If ˛ ¤ ˛1 ; then ˛ D 1 and ˛1 D : Note that P =P 0 Š Zp˛  Zpı Š Zp˛1  Zpı1 : Consequently,

1 D ˛ D ı1 ;

D ˛1 D ı;

which contradicts the statement that P is non-abelian. Thus, ˛ D ˛1 ; D 1 ; and ı D ı1 : (ii) Let b1 D am b n ; where m and n are nonnegative integers. We show that .n; p/ D 1: Assume that pjn: Then pı

ı

b1 D .am b n /p D b p.1C.1Cp

ı /m C:::C.1Cp ı /m.p ı

pı

ı

1/

/ 2 hb p

ıC1

i;

ı

which contradicts the statement that P 0 D hb p i D hb1 i D h.am b n /p i because jK1 =P 0 j D jK=P 0 j D p ı : Therefore, P D ha; bi D ha; b1 i D SK1 and S \ K1 D 1: Theorem 2. Let G be a metacyclic group of odd order and let Y be a normal cyclic subgroup of G with jY j D exp G: Then G=Y is cyclic and G is split. Proof. Assume that G D HN is a Hall decomposition for a metacyclic factorization G D SK;  D .H / D fp 2 .G/W G has a normal Hall p 0 -subgroup g; and  0 D .N /: Then S D S \ H; K D K \ H; S 0 D S \ N; and K 0 D K \ N by virtue of Lemma 5. For p 2 ; let Hp and Yp D hyp i denote the Sylow p-subgroups of H and Y; respectively. Then Yp C Hp and jYp j D exp Hp : It follows from Lemma 9 that Hp is split. Using Lemma 9, we can find xp 2 Hp such that Hp D hxp ihyp i is a split metacyclic factorization. Let Y

x1 D

xp

and

y1 D

p2

Y p2

yp :

Then H D hx1 ihy1 i is a split metacyclic factorization because H is nilpotent. If N D 1; then G D H is split. If N is nontrivial, then it follows from Lemma 7 that N D S 0 K 0 is a split metacyclic factorization because S 0 \ K 0 D 1: We now show that N D S 0 Y 0 is a split metacyclic factorization, where Y 0 is the Hall  0 -subgroup of Y: For any q 2  0 ; let Yq and Nq D Sq Kq denote the Sylow q-subgroups of Y and N; respectively. Note that Yq C Nq and jYq j D exp Nq : We know that Yq is a kernel of Nq by virtue of Theorem 1. This implies that Nq D Sq Yq is a split metacyclic factorization by virtue of the relation Sq \ Kq D 1 and Lemma 9. Thus, N D

Y q2.N /

Nq D

Y q2.N /

Sq Kq D

Y q2.N /

S q Yq D

Y q2.N /

Sq

Y

Yq D S  0 Y 0 :

q2.N /

Further, S 0 \ Y 0 D 1 because Sq \ Yq D 1: Let S 0 D hx2 i and Y 0 D hy2 i: Then S 0  CN .H /; and the relation Y C G implies that G D HN D hx1 ihy1 ihx2 ihy2 i D hx1 ihx2 ihy1 ihy2 i D hx1 x2 ihy1 y2 i D XY; where X D hx1 x2 i and X \ Y D 1:

YAN YANG AND H E - GUO L IU

1632

Theorem 2 may not hold for a group of even order. Example 2. Let G Š Q8  Z3 D hx; b j x 2 D b 6 ; b 12 D 1; b x D b 7 i: Here, jbj D 12 D exp G and hbi C G; but G is not a split group. In Theorem 2, if Y is not normal, then G may not be a split group. Example 3. Let 4

3

5

2

G D ha; b j a3 D b 3 ; b 3 D 1; b a D b 1C3 i: It is obvious that jaj D exp G; but the relation Œb; a D b

2

1 a

b D b 3 2 hai

shows that hai is not normal and G is nonsplit. Lemma 10. If G D SK is a metacyclic factorization of a metacyclic group G of odd order, then exp G D lcm .jS j; jKj/: Proof. Let S D hxi; K D hyi: We show that jx m y n j  lcm .jS j; jKj/ 8x m y n 2 G: Let r D lcm.jSj; jKj/ and t D jKj: Suppose that y x D y  : It is obvious that  r  1 .mod jKj/: Then, for p 2 .G/ and kp r.p/ D r; we have  r D  kp which implies that  mkp

r.p/

r.p/

.mod p t .p/ /;

1

 1 .mod p t.p/ /: Then it follows from Lemma 4 that 1 C  mk C : : : C  mk.p

r.p/

1/

 p r.p/

.mod p t .p/ /:

Thus,

1C

m

C :::

m.r 1/

D

k X1

 mi .1 C  mk C : : : C  mk.p

r.p/

1/

/

i D0

H) p t.p/ j1 C  m C : : : C  m.r H) .x m y n /r D x mr y n.1C

1/

H) t j1 C  m C : : : C  m.r

m C:::C m.r

1/ /

D 1:

1/

O N S PLIT M ETACYCLIC G ROUPS

1633

ˇ Corollary 2. Let G D SK be a metacyclic factorization of a metacyclic group G of odd order and let jKjˇjSj: Then G is split. Example 4. Let G D SK D haihbi be a metacyclic factorization. 3

5

3

2

I. Suppose that G D ha; b j a3 D b 3 ; b 3 D 1; b a D b 1C3 i: Then the relation jKj D 35 D exp G implies that G is split. Let x D ab 24 : Then we can obtain the following representation of G : 2

5

3

G D hx; b j x 3 D 1; b 3 D 1; b a D b 1C3 i: 3

5

3

4

II. Suppose that G D ha; b j a3 D b 3 ; b 3 D 1; b a D b 1C3 i: Then the relations S C G and 3 jS j D 36 D exp G imply that G is split for P 0 D hb 3 i C S: Let x D b 1 a6 : Then we can obtain the following representation of G : 3

6

4

G D hx; a j x 3 D 1; a3 D 1; ax D b 1C3 i: 3

3

4

III. Suppose that G D ha; b j a3 D b 3 7 ; b 3 7 D 1; b a D b 415 i: Then the relation jKj D 34  7 D exp G implies that G is split. Let x D ab 14 : Then we can obtain the following representation of G : 3

G D hx; b j x 3 D 1; b 3

4 7

D 1; b x D b 415 i:

This work was supported by the Natural Science Foundation of the Hubei University of Arts and Sciences (grant No. 2010YA002). REFERENCES 1. 2. 3. 4. 5. 6.

C. E. Hempel, “Metacyclic groups,” Commun. Alg., 28, 3865–3897 (2000). Hyo-Seob Sim, “Metacyclic groups of odd order,” Proc. London Math. Soc., 69, 47–71 (1994). J. N. S. Bidwell and M. J. Curran, “The automorphism group of a split metacyclic p-group,” Arch. Math., 87, 488–497 (2006). M. J. Curran, “The automorphism group of a split metacyclic 2-group,” Arch. Math., 89, 10–23 (2007). M. Golasinski and D. L. Goncalves, “On automorphisms of split metacyclic groups,” Manuscr. Math., 128, 251–273 (2009). M. J. Curran, “The automorphism group of a nonsplit metacyclic p-group,” Arch. Math., 90, 483–489 (2008).