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FINITE GROUPS WITH LARGE CHERMAK–DELGADO LATTICES

Part of: Representation theory of groups

Published online by Cambridge University Press:  25 August 2022

GEORGIANA FASOLĂ Open the ORCID record for GEORGIANA FASOLĂ [Opens in a new window]  and
MARIUS TǍRNǍUCEANU Open the ORCID record for MARIUS TǍRNǍUCEANU [Opens in a new window]
GEORGIANA FASOLĂ
Affiliation:
Faculty of Mathematics, ‘Al.I. Cuza’ University, Iaşi, Romania e-mail: georgiana.fasola@student.uaic.ro
MARIUS TǍRNǍUCEANU*
Affiliation:
Faculty of Mathematics, ‘Al.I. Cuza’ University, Iaşi, Romania
*
e-mail: tarnauc@uaic.ro
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Abstract

Given a finite group G, we denote by $L(G)$ the subgroup lattice of G and by ${\cal CD}(G)$ the Chermak–Delgado lattice of G. In this note, we determine the finite groups G such that $|{\cal CD}(G)|=|L(G)|-k$ , for $k=1,2$ .

Keywords

Chermak–Delgado measureChermak–Delgado latticesubgroup latticegeneralised quaternion 2-group

MSC classification

Primary: 20D30: Series and lattices of subgroups
Secondary: 20D60: Arithmetic and combinatorial problems 20D99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 3 , June 2023 , pp. 451 - 455
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

An, L., Brennan, J. P., Qu, H. and Wilcox, E., ‘Chermak–Delgado lattice extension theorems’, Comm. Algebra 43 (2015), 2201–2213.10.1080/00927872.2014.889147CrossRefGoogle Scholar
Brewster, B., Hauck, P. and Wilcox, E., ‘Groups whose Chermak–Delgado lattice is a chain’, J. Group Theory 17 (2014), 253–279.10.1515/jgt-2013-0043CrossRefGoogle Scholar
Brewster, B. and Wilcox, E., ‘Some groups with computable Chermak–Delgado lattices’, Bull. Aust. Math. Soc. 86 (2012), 29–40.CrossRefGoogle Scholar
Chermak, A. and Delgado, A., ‘A measuring argument for finite groups’, Proc. Amer. Math. Soc. 107 (1989), 907–914.CrossRefGoogle Scholar
Isaacs, I. M., Finite Group Theory (American Mathematical Society, Providence, RI, 2008).Google Scholar
McCulloch, R., ‘Chermak–Delgado simple groups’, Comm. Algebra 45 (2017), 983–991.10.1080/00927872.2016.1172623CrossRefGoogle Scholar
McCulloch, R. and Tărnăuceanu, M., ‘Two classes of finite groups whose Chermak–Delgado lattice is a chain of length zero’, Comm. Algebra 46 (2018), 3092–3096.CrossRefGoogle Scholar
McCulloch, R. and Tărnăuceanu, M., ‘On the Chermak–Delgado lattice of a finite group’, Comm. Algebra 48 (2020), 37–44.CrossRefGoogle Scholar
Schmidt, R., Subgroup Lattices of Groups, de Gruyter Expositions in Mathematics, 14 (de Gruyter, Berlin, 1994).CrossRefGoogle Scholar
Suzuki, M., Group Theory, I, II (Springer-Verlag, Berlin, 1982, 1986).Google Scholar
Tărnăuceanu, M., ‘A note on the Chermak–Delgado lattice of a finite group’, Comm. Algebra 46 (2018), 201–204.10.1080/00927872.2017.1355374CrossRefGoogle Scholar
Tărnăuceanu, M., ‘Finite groups with a certain number of values of the Chermak–Delgado measure’, J. Algebra Appl. 19 (2020), Article no. 2050088.CrossRefGoogle Scholar
Wilcox, E., ‘Exploring the Chermak–Delgado lattice’, Math. Mag. 89 (2016), 38–44.CrossRefGoogle Scholar
Zuccari, A. M., Russo, V. and Scoppola, C. M., ‘The Chermak–Delgado measure in finite $p$ -groups’, J. Algebra 502 (2018), 262–276.10.1016/j.jalgebra.2018.01.030CrossRefGoogle Scholar