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November  2020, 14(4): 603-611. doi: 10.3934/amc.2020033

Designs from maximal subgroups and conjugacy classes of Ree groups

Jamshid Moori 1, , Bernardo G. Rodrigues 2, , Amin Saeidi 1, and  Seiran Zandi 2,,

1. 

School of Mathematical Sciences, North-West University, (Mafikeng) 2754, South Africa
2. 

School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal, Durban 4000, South Africa

* Corresponding author: Seiran Zandi

Received  August 2018 Revised  August 2019 Published  November 2020 Early access  November 2019

Fund Project: The first author acknowledges support of NRF and NWU (Mafikeng).
The second author acknowledges support of NRF through Grant Numbers 95725 and 106071.
The third author acknowledges support of NWU (Mafikeng) postdoctoral fellowship.
The fourth author acknowledges support of NRF postdoctoral fellowship through Grant Number 91495

In this paper, using a method of construction of $ 1 $-designs which are not necessarily symmetric, introduced by Key and Moori in [5], we determine a number of $ 1 $-designs with interesting parameters from the maximal subgroups and the conjugacy classes of the small Ree groups $ ^2G_2(q) $. The designs we obtain are invariant under the action of the groups $ ^2G_2(q) $.

Keywords: Ree groups, design, conjugacy class, maximal subgroup, group action.
Mathematics Subject Classification: 20D05, 05E15, 05E20.
Citation: Jamshid Moori, Bernardo G. Rodrigues, Amin Saeidi, Seiran Zandi. Designs from maximal subgroups and conjugacy classes of Ree groups. Advances in Mathematics of Communications, 2020, 14 (4) : 603-611. doi: 10.3934/amc.2020033
References:
[1]

E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781316529836.

[2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Oxford University Press, Eynsham, 1985. 
[3]

I. M. Isaacs, Character Theory of Finite Groups, Dover Publications, Inc., New York, 1994.

[4]

J. D. Key and J. Moori, Codes, designs and graphs from the Janko groups J1 and J2, J. Combin. Math. Combin. Comput., 40 (2002), 143-159. 

[5]

J. D. Key and J. Moori, Designs from maximal subgroups and conjugacy classes of finite simple groups, J. Combin. Math. Combin. Comput., 99 (2016), 41-60. 

[6]

J. D. Key, J. Moori and B. G. Rodrigues, On some designs and codes from primitive representations of some finite simple groups, Combin. Math. Combin. Comput., 45 (2003), 3-19. 

[7]

O. H. King, The subgroup structure of finite classical groups in terms of geometric configurations, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, 2005, 26-56. doi: 10.1017/CBO9780511734885.003.

[8]

V. M. Levchuk and Y. N. Nuzhin, The structure of Ree groups, Algebra i Logika, 24 (1985), 26-41. 

[9]

J. Moori, Finite groups, designs and codes, in Information Security, Coding Theory and Related Combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 29, IOS, Amsterdam, 2011,202-230.

[10]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly even code invariant under McL:2, J. Combin. Theory Ser. A, 110 (2005), 53-69.  doi: 10.1016/j.jcta.2004.10.001.

[11]

J. Moori and B. G. Rodrigues, Some designs and codes invariant under the simple group Co2, J. Algebra, 316 (2007), 649-661.  doi: 10.1016/j.jalgebra.2007.02.004.

[12]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly-even code invariant under the McL, Ars Combin., 91 (2009), 321-332. 

[13]

J. Moori and B. G. Rodrigues, On some designs and codes invariant under the Higman-Sims group, Util. Math., 86 (2011), 225-239. 

[14]

J. Moori, B. G. Rodrigues, A. Saeidi and S. Zandi, Some symmetric designs invariant under the small Ree groups, Comm. Algebra, 47 (2019), 2131-2148.  doi: 10.1080/00927872.2018.1530245.

[15]

J. Moori and A. Saeidi, Some designs and codes invariant under the Tits group, Adv. Math. Commun., 11 (2017), 77-82.  doi: 10.3934/amc.2017003.

[16]

J. Moori and A. Saeidi, Some design invariant under the Suzuki groups, Util. Math., 109 (2018), 105-114. 

[17]

J. Moori and A. Saeidi, Constructing some design invariant under the PSL2(q), q even, Comm. Algebra, 46 (2018), 160-166.  doi: 10.1080/00927872.2017.1316854.

[18]

R. Ree, A family of simple groups asssociated with the simple Lie algebra of type (G2), Amer. J. Math, 83 (1961), 432-462.  doi: 10.2307/2372888.

[19]

D. O. Revin and E. P. Vdovin, On the number of classes of conjugate Hall subgroups in finite simple groups, J. Algebra, 324 (2010), 3614-3652.  doi: 10.1016/j.jalgebra.2010.09.014.

[20]

H. Ward, On Ree's series of simple groups, Trans. Amer. Math. Soc., 121 (1966), 62-89.  doi: 10.2307/1994333.

[21]

R. A. Wilson, The Finite Simple Groups, Graduate Texts in Mathematics, 251, Springer-Verlag London, Ltd., London, 2009. doi: 10.1007/978-1-84800-988-2.

[22]

R. A. Wilson, Another new approach to the small Ree groups, Arch. Math. (Basel), 94 (2010), 501-510.  doi: 10.1007/s00013-010-0130-4.

show all references

References:
[1]

E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781316529836.

[2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Oxford University Press, Eynsham, 1985. 
[3]

I. M. Isaacs, Character Theory of Finite Groups, Dover Publications, Inc., New York, 1994.

[4]

J. D. Key and J. Moori, Codes, designs and graphs from the Janko groups J1 and J2, J. Combin. Math. Combin. Comput., 40 (2002), 143-159. 

[5]

J. D. Key and J. Moori, Designs from maximal subgroups and conjugacy classes of finite simple groups, J. Combin. Math. Combin. Comput., 99 (2016), 41-60. 

[6]

J. D. Key, J. Moori and B. G. Rodrigues, On some designs and codes from primitive representations of some finite simple groups, Combin. Math. Combin. Comput., 45 (2003), 3-19. 

[7]

O. H. King, The subgroup structure of finite classical groups in terms of geometric configurations, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, 2005, 26-56. doi: 10.1017/CBO9780511734885.003.

[8]

V. M. Levchuk and Y. N. Nuzhin, The structure of Ree groups, Algebra i Logika, 24 (1985), 26-41. 

[9]

J. Moori, Finite groups, designs and codes, in Information Security, Coding Theory and Related Combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 29, IOS, Amsterdam, 2011,202-230.

[10]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly even code invariant under McL:2, J. Combin. Theory Ser. A, 110 (2005), 53-69.  doi: 10.1016/j.jcta.2004.10.001.

[11]

J. Moori and B. G. Rodrigues, Some designs and codes invariant under the simple group Co2, J. Algebra, 316 (2007), 649-661.  doi: 10.1016/j.jalgebra.2007.02.004.

[12]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly-even code invariant under the McL, Ars Combin., 91 (2009), 321-332. 

[13]

J. Moori and B. G. Rodrigues, On some designs and codes invariant under the Higman-Sims group, Util. Math., 86 (2011), 225-239. 

[14]

J. Moori, B. G. Rodrigues, A. Saeidi and S. Zandi, Some symmetric designs invariant under the small Ree groups, Comm. Algebra, 47 (2019), 2131-2148.  doi: 10.1080/00927872.2018.1530245.

[15]

J. Moori and A. Saeidi, Some designs and codes invariant under the Tits group, Adv. Math. Commun., 11 (2017), 77-82.  doi: 10.3934/amc.2017003.

[16]

J. Moori and A. Saeidi, Some design invariant under the Suzuki groups, Util. Math., 109 (2018), 105-114. 

[17]

J. Moori and A. Saeidi, Constructing some design invariant under the PSL2(q), q even, Comm. Algebra, 46 (2018), 160-166.  doi: 10.1080/00927872.2017.1316854.

[18]

R. Ree, A family of simple groups asssociated with the simple Lie algebra of type (G2), Amer. J. Math, 83 (1961), 432-462.  doi: 10.2307/2372888.

[19]

D. O. Revin and E. P. Vdovin, On the number of classes of conjugate Hall subgroups in finite simple groups, J. Algebra, 324 (2010), 3614-3652.  doi: 10.1016/j.jalgebra.2010.09.014.

[20]

H. Ward, On Ree's series of simple groups, Trans. Amer. Math. Soc., 121 (1966), 62-89.  doi: 10.2307/1994333.

[21]

R. A. Wilson, The Finite Simple Groups, Graduate Texts in Mathematics, 251, Springer-Verlag London, Ltd., London, 2009. doi: 10.1007/978-1-84800-988-2.

[22]

R. A. Wilson, Another new approach to the small Ree groups, Arch. Math. (Basel), 94 (2010), 501-510.  doi: 10.1007/s00013-010-0130-4.

Table 1.  Non-trivial designs from $G = Ree(q)$ using construction Method 2
$Max$ $t =o(x)$ $v = |x^G|$ $k =| M \cap x^G|$ $ \lambda= \chi_{M_i}(x)$
$M_1$ $t=2$ $q^2(q^2-q+1)$ $q^2$ $q+1$
$M_1$ $t=3$ $(q^3+1)(q-1)$ $q-1$ 1
$M_1$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q(q-1)}{2}$ 1
$M_1$ $t=9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q^2(q-1)}{3}$ 1
$M_1$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2 (q-1)}{2}$ 1
$M_1$ $t |(q-1)$, $t \ne 2$ ${q^3}(q^3+1)$ $2q^3$ 2
$M_2, M_3$ $ t=2$ $q^2(q^2-q+1) $ $ q^{\mp}$ $\frac{{{q(q^2-1)}}}{6}$
$M_2, M_3$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $ q^{\mp}$ $\frac{q^2}{3}$
$M_2, M_3$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $ q^{\mp}$ $\frac{q}{3}$
$M_2, M_3$ $t | q^{\mp}$ ${q^3(q^2-1)q^{\pm}}$ $ 6$ $1$
$M_4$ $ t=2$ $q^2(q^2-q+1) $ $ q^2-q+1$ $q^2-q+1$
$M_4$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $ \frac{q^2-1}{2}$ $q$
$M_4$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2} $ $ \frac{q^2-1}{2}$ $1$
$M_4$ $t |(q-1)$, $t \ne 2$ $q^3(q^3+1)$ $ q(q+1)$ $1$
$M_4$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $ 3q(q-1))$ $3$
$M_5$ $t=2$ $q^2(q^2-q+1) $ $ q+4$ $\frac{{{q(q-1)(q+4)}}}{6}$
$M_5$ $ t=3$ $\frac{{{{q}(q^3+1)(q-1)}}}{2}$ $ q+1$ $ \frac{{{q^2}}}{3}$
$M_5$ $ t=6$ $\frac{{{{q^2}(q^3+1)(q-1)}}}{2}$ $ q+1$ $ \frac{{{q}}}{3}$
$M_5$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $ 6$ 1
$M_6$ $t=2$ $q^2(q^2-q+1) $ $ q_0^2(q_0^2-q_0+1)$ $\frac{q(q^2-1)}{q_0(q_0^2-1)}$
$M_6$ $ t=3$ $(q^3+1)(q-1)$ $(q_0^3+1)(q_0-1)$ $ \frac{{{q^3}}}{q_0^3}$
$M_6$ $ t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q_0(q_0^3+1)(q_0-1)}{2}$ $ \frac{{{q^2}}}{q_0^2}$
$M_6$ $t= 9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{3}$ $ \frac{{{q}}}{q_0}$
$M_6$ $t= 6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{2}$ $ \frac{{{q}}}{q_0}$
$M_6$ $t |(q_0-1)$, $t \ne 2$ ${q^3(q^3+1)}$ ${q_0^3(q_0^3+1)}$ $ \frac{q-1}{q_0-1} $
$M_6$ $t |\frac{q_0+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ ${q_0^3(q_0^2-q_0+1)(q_0-1)}$ $ \frac{q+1}{q_0+1} $
$^*M_6$ $t|q_0^{\pm}$ ${q^3}\left( {q^2 - 1} \right){q^{\pm}} $ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}} $ $ \frac{{q^{\mp}}}{{q_0^{\mp}}} $
$^{**}M_6$ $t|q_0^{\pm}$ ${q^3(q^3+1)}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}} $ $\frac{q-1}{q_0^{\mp}}$
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$Max$ $t =o(x)$ $v = |x^G|$ $k =| M \cap x^G|$ $ \lambda= \chi_{M_i}(x)$
$M_1$ $t=2$ $q^2(q^2-q+1)$ $q^2$ $q+1$
$M_1$ $t=3$ $(q^3+1)(q-1)$ $q-1$ 1
$M_1$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q(q-1)}{2}$ 1
$M_1$ $t=9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q^2(q-1)}{3}$ 1
$M_1$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2 (q-1)}{2}$ 1
$M_1$ $t |(q-1)$, $t \ne 2$ ${q^3}(q^3+1)$ $2q^3$ 2
$M_2, M_3$ $ t=2$ $q^2(q^2-q+1) $ $ q^{\mp}$ $\frac{{{q(q^2-1)}}}{6}$
$M_2, M_3$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $ q^{\mp}$ $\frac{q^2}{3}$
$M_2, M_3$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $ q^{\mp}$ $\frac{q}{3}$
$M_2, M_3$ $t | q^{\mp}$ ${q^3(q^2-1)q^{\pm}}$ $ 6$ $1$
$M_4$ $ t=2$ $q^2(q^2-q+1) $ $ q^2-q+1$ $q^2-q+1$
$M_4$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $ \frac{q^2-1}{2}$ $q$
$M_4$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2} $ $ \frac{q^2-1}{2}$ $1$
$M_4$ $t |(q-1)$, $t \ne 2$ $q^3(q^3+1)$ $ q(q+1)$ $1$
$M_4$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $ 3q(q-1))$ $3$
$M_5$ $t=2$ $q^2(q^2-q+1) $ $ q+4$ $\frac{{{q(q-1)(q+4)}}}{6}$
$M_5$ $ t=3$ $\frac{{{{q}(q^3+1)(q-1)}}}{2}$ $ q+1$ $ \frac{{{q^2}}}{3}$
$M_5$ $ t=6$ $\frac{{{{q^2}(q^3+1)(q-1)}}}{2}$ $ q+1$ $ \frac{{{q}}}{3}$
$M_5$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $ 6$ 1
$M_6$ $t=2$ $q^2(q^2-q+1) $ $ q_0^2(q_0^2-q_0+1)$ $\frac{q(q^2-1)}{q_0(q_0^2-1)}$
$M_6$ $ t=3$ $(q^3+1)(q-1)$ $(q_0^3+1)(q_0-1)$ $ \frac{{{q^3}}}{q_0^3}$
$M_6$ $ t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q_0(q_0^3+1)(q_0-1)}{2}$ $ \frac{{{q^2}}}{q_0^2}$
$M_6$ $t= 9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{3}$ $ \frac{{{q}}}{q_0}$
$M_6$ $t= 6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{2}$ $ \frac{{{q}}}{q_0}$
$M_6$ $t |(q_0-1)$, $t \ne 2$ ${q^3(q^3+1)}$ ${q_0^3(q_0^3+1)}$ $ \frac{q-1}{q_0-1} $
$M_6$ $t |\frac{q_0+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ ${q_0^3(q_0^2-q_0+1)(q_0-1)}$ $ \frac{q+1}{q_0+1} $
$^*M_6$ $t|q_0^{\pm}$ ${q^3}\left( {q^2 - 1} \right){q^{\pm}} $ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}} $ $ \frac{{q^{\mp}}}{{q_0^{\mp}}} $
$^{**}M_6$ $t|q_0^{\pm}$ ${q^3(q^3+1)}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}} $ $\frac{q-1}{q_0^{\mp}}$
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