Finite groups having exactly 28 elements of maximal order

Han, Zhangjia; Song, Ren

doi:10.12988/ija.2014.4668

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G| ≤1

Finite Groups With Few Elements Of Maximal Order1

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2021

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“…. (see [2,[4][5][6][7][8][9][10]14]). There are exactly 13007 groups with k < 36, the largest have order 3264.…”

Section: G| ≤mentioning

confidence: 99%

On a bound of Cocke and Venkataraman

Sambale

Wellmann

2021

Monatsh Math

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Let G be a finite group with exactly k elements of largest possible order m. Let q(m) be the product of $$\gcd (m,4)$$ gcd ( m , 4 ) and the odd prime divisors of m. We show that $$|G|\le q(m)k^2/\varphi (m)$$ | G | ≤ q ( m ) k 2 / φ ( m ) where $$\varphi $$ φ denotes Euler’s totient function. This strengthens a recent result of Cocke and Venkataraman. As an application we classify all finite groups with $$k<36$$ k < 36 . This is motivated by a conjecture of Thompson and unifies several partial results in the literature.

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“…. (see [2,[4][5][6][7][8][9][10]14]). There are exactly 13007 groups with k < 36, the largest have order 3264.…”

Section: G| ≤mentioning

confidence: 99%

On a bound of Cocke and Venkataraman

Sambale

Wellmann

2021

Monatsh Math

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show abstract

“…Suppose that k < 36. First we consider |G| = 2 9 . Any element x ∈ G of maximal order m must lie in some maximal subgroup M < G (otherwise G is cyclic and k = 2 8 > 35).…”

Section: Finite Groups With Few Elements Of Maximal Ordermentioning

confidence: 99%

On a bound of Cocke and Venkataraman

Sambale¹,

Wellmann²

2021

Preprint

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Let G be a finite group with exactly k elements of largest possible order m. Let q(m) be the product of gcd(m, 4) and the odd prime divisors of m. We show that |G| ≤ q(m)k 2 /ϕ(m) where ϕ denotes Euler's totient function. This strengthens a recent result of Cocke and Venkataraman. As an application we classify all finite groups with k < 36. This is motivated by a conjecture of Thompson and unifies several partial results in the literature.

show abstract

Finite groups having exactly 28 elements of maximal order

Cited by 2 publications

References 0 publications

On a bound of Cocke and Venkataraman

On a bound of Cocke and Venkataraman

On a bound of Cocke and Venkataraman

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