The Chermak–Delgado measure of a subgroup of a finite group is defined as . The subgroups with maximal Chermak–Delgado measure form a poset and corresponding lattice, known as the CD-lattice of . We describe the symmetric nature of CD-lattices in general, and use information about centrally large subgroups to determine the CD-lattices of split metacyclic -groups in particular. We also describe a rank-symmetric sublattice of the CD-lattice of split metacyclic -groups.
References
L. An, J. P. Brennan, H. Qu, and E. Wilcox, “Chermak–Delgado lattice extension theorems”, Comm. Algebra 43:5 (2015), 2201–2213. MR3316845 1327.20020 10.1080/00927872.2014.889147 L. An, J. P. Brennan, H. Qu, and E. Wilcox, “Chermak–Delgado lattice extension theorems”, Comm. Algebra 43:5 (2015), 2201–2213. MR3316845 1327.20020 10.1080/00927872.2014.889147
Y. Berkovich, “Maximal abelian and minimal nonabelian subgroups of some finite two-generator $p$-groups especially metacyclic”, Israel J. Math. 194:2 (2013), 831–869. MR3047093 1278.20019 10.1007/s11856-012-0121-1 Y. Berkovich, “Maximal abelian and minimal nonabelian subgroups of some finite two-generator $p$-groups especially metacyclic”, Israel J. Math. 194:2 (2013), 831–869. MR3047093 1278.20019 10.1007/s11856-012-0121-1
J. N. S. Bidwell and M. J. Curran, “Automorphisms of finite abelian groups”, Math. Proc. R. Ir. Acad. 110A:1 (2010), 57–71. MR2666671 1275.20018 J. N. S. Bidwell and M. J. Curran, “Automorphisms of finite abelian groups”, Math. Proc. R. Ir. Acad. 110A:1 (2010), 57–71. MR2666671 1275.20018
B. Brewster and E. Wilcox, “Some groups with computable Chermak–Delgado lattices”, Bull. Aust. Math. Soc. 86:1 (2012), 29–40. MR2960225 1259.20022 10.1017/S0004972712000196 B. Brewster and E. Wilcox, “Some groups with computable Chermak–Delgado lattices”, Bull. Aust. Math. Soc. 86:1 (2012), 29–40. MR2960225 1259.20022 10.1017/S0004972712000196
B. Brewster, P. Hauck, and E. Wilcox, “Groups whose Chermak–Delgado lattice is a chain”, J. Group Theory 17:2 (2014), 253–265. MR3176659 1308.20018 10.1515/jgt-2013-0043 B. Brewster, P. Hauck, and E. Wilcox, “Groups whose Chermak–Delgado lattice is a chain”, J. Group Theory 17:2 (2014), 253–265. MR3176659 1308.20018 10.1515/jgt-2013-0043
B. Brewster, P. Hauck, and E. Wilcox, “Quasi-antichain Chermak–Delgado lattices of finite groups”, Arch. Math. $($Basel$)$ 103:4 (2014), 301–311. MR3274815 1308.20019 10.1007/s00013-014-0696-3 B. Brewster, P. Hauck, and E. Wilcox, “Quasi-antichain Chermak–Delgado lattices of finite groups”, Arch. Math. $($Basel$)$ 103:4 (2014), 301–311. MR3274815 1308.20019 10.1007/s00013-014-0696-3
A. Chermak and A. Delgado, “A measuring argument for finite groups”, Proc. Amer. Math. Soc. 107:4 (1989), 907–914. MR994774 0687.20022 10.1090/S0002-9939-1989-0994774-4 A. Chermak and A. Delgado, “A measuring argument for finite groups”, Proc. Amer. Math. Soc. 107:4 (1989), 907–914. MR994774 0687.20022 10.1090/S0002-9939-1989-0994774-4
R. M. Davitt, “The automorphism group of a finite metacyclic $p$-group”, Proc. Amer. Math. Soc. 25 (1970), 876–879. MR0285594 0202.02501 R. M. Davitt, “The automorphism group of a finite metacyclic $p$-group”, Proc. Amer. Math. Soc. 25 (1970), 876–879. MR0285594 0202.02501
J. Dietz, “Stable splittings of classifying spaces of metacyclic $p$-groups, $p$ odd”, J. Pure Appl. Algebra 90:2 (1993), 115–136. MR1250764 0790.55011 10.1016/0022-4049(93)90125-D J. Dietz, “Stable splittings of classifying spaces of metacyclic $p$-groups, $p$ odd”, J. Pure Appl. Algebra 90:2 (1993), 115–136. MR1250764 0790.55011 10.1016/0022-4049(93)90125-D
G. Glauberman, “Centrally large subgroups of finite $p$-groups”, J. Algebra 300:2 (2006), 480–508. MR2228208 1103.20013 10.1016/j.jalgebra.2005.11.033 G. Glauberman, “Centrally large subgroups of finite $p$-groups”, J. Algebra 300:2 (2006), 480–508. MR2228208 1103.20013 10.1016/j.jalgebra.2005.11.033
L. Héthelyi and B. Külshammer, “Characters, conjugacy classes and centrally large subgroups of $p$-groups of small rank”, J. Algebra 340 (2011), 199–210. MR2813569 1244.20005 10.1016/j.jalgebra.2011.05.026 L. Héthelyi and B. Külshammer, “Characters, conjugacy classes and centrally large subgroups of $p$-groups of small rank”, J. Algebra 340 (2011), 199–210. MR2813569 1244.20005 10.1016/j.jalgebra.2011.05.026
I. M. Isaacs, Finite group theory, Graduate Studies in Mathematics 92, American Mathematical Society, Providence, RI, 2008. MR2426855 1169.20001 I. M. Isaacs, Finite group theory, Graduate Studies in Mathematics 92, American Mathematical Society, Providence, RI, 2008. MR2426855 1169.20001
B. W. King, “Presentations of metacyclic groups”, Bull. Austral. Math. Soc. 8 (1973), 103–131. MR0323893 0245.20016 10.1017/S0004972700045500 B. W. King, “Presentations of metacyclic groups”, Bull. Austral. Math. Soc. 8 (1973), 103–131. MR0323893 0245.20016 10.1017/S0004972700045500
A. Mann, “The number of subgroups of metacyclic groups”, pp. 93–95 in Character theory of finite groups, Contemp. Math. 524, Amer. Math. Soc., Providence, RI, 2010. MR2731922 1213.20015 A. Mann, “The number of subgroups of metacyclic groups”, pp. 93–95 in Character theory of finite groups, Contemp. Math. 524, Amer. Math. Soc., Providence, RI, 2010. MR2731922 1213.20015
