# Periodic subgroups of projective linear groups in positive characteristic

Open Mathematics (2008)

• Volume: 6, Issue: 3, page 384-392
• ISSN: 2391-5455

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## Abstract

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We classify the maximal irreducible periodic subgroups of PGL(q, $𝔽$ ), where $𝔽$ is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and $𝔽$ × has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, $𝔽$ ) containing the centre $𝔽$ ×1q of GL(q, $𝔽$ ), such that G/$𝔽$ ×1q is a maximal periodic subgroup of PGL(q, $𝔽$ ), and if H is another group of this kind then H is GL(q, $𝔽$ )-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q, $𝔽$ ) is self-normalising.

## How to cite

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Alla Detinko, and Dane Flannery. "Periodic subgroups of projective linear groups in positive characteristic." Open Mathematics 6.3 (2008): 384-392. <http://eudml.org/doc/269275>.

abstract = {We classify the maximal irreducible periodic subgroups of PGL(q, $\mathbb \{F\}$ ), where $\mathbb \{F\}$ is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and $\mathbb \{F\}$ × has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, $\mathbb \{F\}$ ) containing the centre $\mathbb \{F\}$ ×1q of GL(q, $\mathbb \{F\}$ ), such that G/$\mathbb \{F\}$ ×1q is a maximal periodic subgroup of PGL(q, $\mathbb \{F\}$ ), and if H is another group of this kind then H is GL(q, $\mathbb \{F\}$ )-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q, $\mathbb \{F\}$ ) is self-normalising.},
author = {Alla Detinko, Dane Flannery},
journal = {Open Mathematics},
keywords = {linear group; periodic group; projective general linear group; field; classification; linear groups over finite fields; maximal irreducible periodic subgroups},
language = {eng},
number = {3},
pages = {384-392},
title = {Periodic subgroups of projective linear groups in positive characteristic},
url = {http://eudml.org/doc/269275},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Alla Detinko
AU - Dane Flannery
TI - Periodic subgroups of projective linear groups in positive characteristic
JO - Open Mathematics
PY - 2008
VL - 6
IS - 3
SP - 384
EP - 392
AB - We classify the maximal irreducible periodic subgroups of PGL(q, $\mathbb {F}$ ), where $\mathbb {F}$ is a field of positive characteristic p transcendental over its prime subfield, q = p is prime, and $\mathbb {F}$ × has an element of order q. That is, we construct a list of irreducible subgroups G of GL(q, $\mathbb {F}$ ) containing the centre $\mathbb {F}$ ×1q of GL(q, $\mathbb {F}$ ), such that G/$\mathbb {F}$ ×1q is a maximal periodic subgroup of PGL(q, $\mathbb {F}$ ), and if H is another group of this kind then H is GL(q, $\mathbb {F}$ )-conjugate to a group in the list. We give criteria for determining when two listed groups are conjugate, and show that a maximal irreducible periodic subgroup of PGL(q, $\mathbb {F}$ ) is self-normalising.
LA - eng
KW - linear group; periodic group; projective general linear group; field; classification; linear groups over finite fields; maximal irreducible periodic subgroups
UR - http://eudml.org/doc/269275
ER -

## References

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12. [12] Mazurova V.N., Periodic subgroups of classical groups over fields of positive characteristic, Dokl. Akad. Nauk BSSR, 1985, 29, 493–496 (in Russian) Zbl0571.20042
13. [13] Suprunenko D.A., Matrix groups, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1976, 45
14. [14] Wehrfritz B.A.F., Infinite linear groups, Springer-Verlag, Berlin, Heidelberg, New York, 1973 Zbl0261.20038
15. [15] Winter D.J., Representations of locally finite groups, Bull. Amer. Math. Soc., 1968, 74, 145–148 http://dx.doi.org/10.1090/S0002-9904-1968-11913-5 Zbl0159.31304
16. [16] Zalesskii A.E., Maximal periodic subgroups of the full linear group over a field with positive characteristic, Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk, 1966, 2, 121–123 (in Russian)
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