group

Etymology

From French groupe (“cluster, group”), from Italian gruppo, groppo (“a knot, heap, group, bag (of money)”), from Vulgar Latin *cruppo, Renaissance Latin grupus, from Frankish *krupp, from Proto-Germanic *kruppaz (“lump, round mass, body, crop”), from Proto-Indo-European *grewb- (“to crumple, bend, crawl”). Cognate with German Kropf (“crop, craw, bunch”), Old English cropp, croppa (“cluster, bunch, sprout, flower, berry, ear of corn, crop”), Dutch krop (“craw”), Icelandic kroppr (“hump, bunch”). Doublet of crop and croup.

noun

1. A number of things or persons being in some relation to one another.
   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. 2013-07-19, Peter Wilby, “Finland spreads word on schools”, in The Guardian Weekly, volume 189, number 6, page 30
   there is a group of houses behind the hill; he left town to join a Communist group
   A group of people gathered in front of the Parliament to demonstrate against the Prime Minister's proposals.
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.
   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
   2007, Zhong-Qi Ma, Group Theory for Physicists, World Scientific, page 277, In Chap. 4 the fundamental concepts on Lie groups have been introduced through the SO(3) group and its covering group SU(2).
3. (geometry, archaic) An effective divisor on a curve.
4. A (usually small) group of people who perform music together.
   Did you see the new jazz group?
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.
   Nitro is an electron-withdrawing 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.
    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. 2012-10-15, Sid Lowe, “Spain aim to take 'very big step' towards 2014 World Cup against France”, in The Guardian
15. (business) A commercial organization.

verb

1. (transitive) To put together to form a group.
   group the dogs by hair colour
2. (intransitive) To come together to form a group.
   For many people forming pods last year, finding compatible people to group with was not a cost but a goal. 2021-10-01, Calder Katyal, “Schools Need to Undo the Damage of Pods”, in The Atlantic