Divisible group for a set of primes
From Groupprops
Contents
Definition
Let be a set of primes. A group
is termed
-divisible if it satisfies the following equivalent definitions:
No. | Shorthand | Explanation |
---|---|---|
1 | ![]() ![]() |
For every ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
2 | ![]() ![]() ![]() |
if ![]() ![]() ![]() ![]() ![]() |
3 | (not necessarily unique) rational powers with denominators ![]() |
if ![]() ![]() ![]() ![]() ![]() ![]() |
Related notions
- Powered group for a set of primes
- Powering-injective group for a set of primes
- Torsion-free group for a set of primes
Conjunction with group properties
- Nilpotent group that is divisible for a set of primes is the conjunction with the property of being a nilpotent group.
References
Journal references
- Some aspects of groups with unique roots by Gilbert Baumslag, Acta mathematica, Volume 104, Page 217 - 303(Year 1960): PDF (ungated)More info: This paper uses the notation
-group to describe what we call a
-divisible group. The notation is introduced in Section 2, Page 218 (second page of the paper).
Textbook references
- p-automorphisms of finite p-groups by Evgenii I. Khukhro, 13-digit ISBN 978-0-521-59717-3, 10-digit ISBN 0-521-59717-X, Page 18, Section 1.3 (Algebraic systems, varieties, and free objects), More info