Definition of "group"

group

/ɡɹuːp/

noun
plural groups
1. A number of things or persons being in some relation to one another.
  Quotations
  Then everybody once more knelt, and soon the blessing was pronounced. The choir and the clergy trooped out slowly, […] , down the nave to the western door. […] At a seemingly immense distance the surpliced group stopped to say the last prayer.
  1918, W[illiam] B[abington] Maxwell, chapter V, in The Mirror and the Lamp, Indianapolis, Ind.: The Bobbs-Merrill Company
  Imagine a country where children do nothing but play until they start compulsory schooling at age seven. Then, without exception, they attend comprehensives until the age of 16. Charging school fees is illegal, and so is sorting pupils into ability groups by streaming or setting.
  2013 July 19, Peter Wilby, “Finland spreads word on schools”, in The Guardian Weekly, volume 189, number 6, page 30
2. (group theory) A set with an associative binary operation, under which there exists an identity element, and such that each element has an inverse.
  Quotations
  Throughout this section, we shall assume the existence of finitely presented groups with unsolvable word problem.
  1977, Roger C. Lyndon, Paul E. Schupp, Combinatorial Group Theory, Springer, page 192
  In this chapter we give some examples of Fuchsian groups. The most interesting and important ones are the so-called "arithmetic" Fuchsian groups, i.e., discrete subgroups of PSL(2,R) obtained by some "arithmetic" operations. One such construction we have already seen: if we choose all matrices of SL(2,R) with integer coefficients, then the corresponding elements of PSL(2,R) form the modular group PSL(2,Z).
  1992, Svetlana Katok, Fuchsian Groups, University of Chicago Press, page 112
  In Chap. 4 the fundamental concepts on Lie groups have been introduced through the SO(3) group and its covering group SU(2).
  2007, Zhong-Qi Ma, Group Theory for Physicists, World Scientific, page 277
3. (geometry, archaic) An effective divisor on a curve.
4. A (usually small) group of people who perform music together.
5. (astronomy) A small number (up to about fifty) of galaxies that are near each other.
6. (chemistry) A column in the periodic table of chemical elements.
7. (chemistry) A functional group.
8. (sociology) A subset of a culture or of a society.
9. (military) An air force formation.
10. (geology) A collection of formations or rock strata.
11. (computing) A number of users with the same rights with respect to accession, modification, and execution of files, computers and peripherals.
12. An element of an espresso machine from which hot water pours into the portafilter.
13. (music) A number of eighth, sixteenth, etc., notes joined at the stems; sometimes rather indefinitely applied to any ornament made up of a few short notes.
14. (sports) A set of teams playing each other in the same division, while not during the same period playing any teams that belong to other sets in the division.
  Quotations
  It is the third of eight matches that Spain will play in Group I, but the coach Vicente del Bosque has described it as being more akin to the first leg of a cup semi-final.
  2012 October 15, Sid Lowe, “Spain aim to take 'very big step' towards 2014 World Cup against France”, in The Guardian
15. (business) A commercial organization.
verb
third-person singular simple present groups, present participle grouping, simple past and past participle grouped
1. (transitive) To put together to form a group.
2. (intransitive) To come together to form a group.
  Quotations
  For many people forming pods last year, finding compatible people to group with was not a cost but a goal.
  2021 October 1, Calder Katyal, “Schools Need to Undo the Damage of Pods”, in The Atlantic‎