Glasnik Matematicki, Vol. 42, No.2 (2007), 345-355.

CYCLIC SUBGROUPS OF ORDER 4 IN FINITE 2-GROUPS

Zvonimir Janko

Abstract.   We determine completely the structure of finite 2-groups which possess exactly six cyclic subgroups of order 4. This is an exceptional case because in a finite 2-group is the number of cyclic subgroups of a given order 2ⁿ (n ≥ 2 fixed) divisible by 4 in most cases and this solves a part of a problem stated by Berkovich. In addition, we show that if in a finite 2-group G all cyclic subgroups of order $4$ are conjugate, then G is cyclic or dihedral. This solves a problem stated by Berkovich.

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Finite 2-groups, 2-groups of maximal class, minimal nonabelian 2-groups, L₂-groups, U₂-groups.

DOI: 10.3336/gm.42.2.08

References:

1. Y. Berkovich, On the number of elements of given order in a finite p-group, Israel J. Math. 73 (1991), 107-112.
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2. Y. Berkovich, Groups of prime power order, Parts I, II, and III (with Z. Janko), in preparation.

3. Z. Janko, Finite 2-groups with exactly four cyclic subgroups of order 2ⁿ, J. reine angew. Math. 566 (2004), 135-181.
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4. Z. Janko, Finite 2-groups G with |Ω₂(G)| = 16, Glas. Mat. Ser. III 40 (2005), 71-86.
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5. Z. Janko, Elements of order at most 4 in finite 2-groups, 2, J. Group Theory 8 (2005), 683-686.
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6. Z. Janko, Finite 2-groups with exactly one nonmetacyclic maximal subgroup, submitted.