Subgroup

In mathematics, given a group G under an operation *, we say that some subset H of G is a subgroup if H is a group under * also. (The same definition applies more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups.)

It is easily shown that H is a subgroup of the group G if and only if it is nonempty and closed to products and inverses. Furthermore, H's identity element is equal to G's identity element, and the inverse of an element of H is the same as the inverse of that element in G.

The subgroups of any given group form a complete lattice under inclusion. There is a minimal subgroup, the trivial group {e} (e being G's identity element), and a maximal subgroup, the group G itself.

If S is a subset of G, then there exists a minimal subgroup containing S; it is denoted by <S> and is said to be generated by S. The elements of <S> are all finite products of elements of S and their inverses. Groups generated by a single element are called cyclic and are isomorphic to either (Z, +), where Z denotes the integers, or to (Z_n, +), where Z_n denotes the integers modulo n for some positive integer n (see modular arithmetic).

Order of an element of a group: Given an element x of G, the order of the cyclic subgroup is called the order of x; it is the smallest positive integer n such that xⁿ = e.

Given a subgroup H and some g in G, we define the left coset g*H = {g*h : h in H}. Because g is invertible, the set g*H has just as many elements as H. Furthermore, every element of G is contained in precisely one left coset of H; the left cosets are the equivalence classes corresponding to the equivalence relation g₁ ~ g₂ iff g₁^-1 * g₂ is in H. The number of left cosets of H is called the index of H in G and is denoted by [G : H]. Lagrange's theorem states that

[G : H] |H| = |G|

where |G| and |H| denote the cardinalities of G and H, respectively. In particular, if G is finite, then the cardinality of every subgroup of G (and the order of every element of G) must be a divisor of |G|.

Right cosets are defined analogously: H*g = {h*g : h in H}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H]. If g*H = H*g for every g in G, then H is said to be a normal subgroup. In that case we can define a multiplication on cosets by

(g1*H)*(g2*H) := (g1*g2)*H

This turns the set of cosets in a group called the quotient group G/H. There is a natural homomorphism f : G -> G/H given by f(g)=g*H. The image f(H) consists only of the identity element of G/H, the coset e*H.

In general, a group homomorphism f: G -> K sends subgroups of G to subgroups of K. Also, the preimage of any subgroup of K is a subgroup of G. We call the preimage of the trivial group {e} in K the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image f(G) of G is always isomorphic to G/ker(f).

The normal subgroups of any group G form a lattice under inclusion. The minimal and maximal elements are {e} and G, the greatest lower bound of two subgroup is their intersection and their least upper bound is a product group.

Commercial Usage: Subgroup

Domain	Title
Books	Dance as ritual drama and entertainment in the Gòáelòáedòâe of the Kâetu-Yoráubâa subgroup in West Africa (reference) Data base management systems, their use in hospital data processing: report of the Data Base Subgroup to the National Health Computing Services Group Working Party on Software and Programming (reference) Generalized Whittaker Functions on $SU(2,2)$ with Respect to the Siegel Parabolic Subgroup (reference) Recent and Fossil Clupemorph Fishes With Materials for Revision of the Subgroup Clupeoids (Bulletin of the American Museum of Natural History, Vol 18) (reference) Subgroup Complexes (reference) (more book examples)
Source: compiled by the editor from various references; see credits.

Non-Fiction Usage: Subgroup

Subject	Topic	Quote
Health		However, the size of this subgroup was not large enough to make the result conclusive. (references)
		The consensus panel agrees that there may be a subgroup of patients with primary HPT that can be safely followed. (references)
		A subgroup analysis in the first randomized trial suggested that antenatal corticosteroid administration might predispose to fetal death in hypertensive women. (references)
Economic History	Philippines	Processed fruit and vegetables is the largest subgroup in this category. (references)
	Cote d'Ivoire	The Baoules, in the Akan division, probably comprise the largest-single subgroup with 15%-20% of the population. (references)
Minorities	Central African Republic	Until 1993 members of Kolingba's ethnic group, the Yakoma subgroup of the Ngbandi, held a disproportionate number of senior positions in government, the armed forces, and state-owned firms. (references)
Source: compiled by the editor from ICON Group International, Inc.; see credits.

Usage Frequency: Subgroup

Expression: Subgroup

Frequency of Internet Keywords: Subgroup

Modern Translation: Subgroup

Language		Translations for "subgroup"; alternative meanings/domain in parentheses.
Albanian		nëngrup. (various references)

Arabic		‏العشيرة في تصنيف الأحياء. (various references)

Bulgarian		подгрупа (subclass). (various references)

Chinese		小群. (various references)

Czech		menší skupina. (various references)

Danish		undergruppe. (various references)

Dutch		subgroep. (various references)

Finnish		esiryhmä. (various references)

French		sous-groupe. (various references)

German		untergruppe. (various references)

Greek		μήτρια εισερχόμενη υποομάδα (matrix inbound subgroup), εισερχόμενη υποομάδα μέσω μήτρας (matrix inbound subgroup). (various references)

Hungarian		alcsoport (subbranch, subfamily, sub-group). (various references)

Italian		sottogruppo. (various references)

Japanese Kanji		亜群 , 亜族 , 亜属 (subgenus). (various references)

Japanese Katakana		あぞく (subgenus), あぐ�". (various references)

Pig Latin		ubgroupsay

Portuguese		subgrupo. (various references)

Romanian		subgrupã. (various references)

Russian		подгруппа. (various references)

Serbo-Croatian		podvrsta (variety), podgrupa. (various references)

Spanish		subgrupo. (various references)

Swedish		undergrupp. (various references)

Thai		กลุ่มย่อย (subclass). (various references)

Ukrainian		підгрупа. (various references)
Source: compiled by the editor from various translation references.

Derivations & Misspellings: Subgroup

Rhyming with "Subgroup"

Anagrams: Subgroup

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "b-g-o-p-r-s-u-u"
	-2 letters: bourgs, groups, rugous.
	-3 letters: bogus, bourg, burgs, burps, gorps, group, grubs, gurus, pours, progs, roups, rubus, sprug, usurp.
	-4 letters: bogs, bops, bros, bugs, burg, burp, burs, gobs, gorp, grub, guru, opus, orbs, ours, pour, prog, pros, pubs, pugs, purs, robs, roup, rubs, rugs, sorb, soup, sour, spur, urbs, urus.
	-5 letters: bog, bop, bos, bro.
	Words containing the letters "b-g-o-p-r-s-u-u"
	+1 letter: subgroups.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Parts of Speech	Percent	Usage per 100 Million Words	Rank in English
Noun (singular)	98.33%	118	29,674
Lexical Verb (infinitive)	0.83%	1	339,140
Lexical Verb (base form)	0.83%	1	339,140
Total	100.00%	120	N/A

Derivations
	Words beginning with "subgroup": subgroups. (additional references)
Misspellings
	"Subgroup" is suggested in spellcheckers for the following: tubigrup. (additional references)
Source: compiled by the editor, based on several corpora (additional references).

# of Phoneme Matches	Pronunciation	Word(s) rhyming with "subgroup" (pronounced su"bgruw'p)
4	-g r uw' p	newsgroup.
3	-r uw' p	paratroop.
Source: compiled by the editor (additional references); see credits.

Expression using "subgroup": matrix inbound subgroup. Additional references.
Source: compiled by the editor from various references; see credits.

Expression	Frequency per Day
subgroup	4
Source: compiled by the editor from various references; see credits.

Subgroup

Definition: Subgroup

Subgroup

Noun

Crosswords: Subgroup

Specialty Definition: Subgroup

Commercial Usage: Subgroup

Books

Non-Fiction Usage: Subgroup

Health

Economic History

Philippines

Cote d'Ivoire

Minorities

Central African Republic

Usage Frequency: Subgroup

Expression: Subgroup

Frequency of Internet Keywords: Subgroup

subgroup

Modern Translation: Subgroup

Albanian

Arabic

Bulgarian

Chinese

Czech

Danish

Dutch

Finnish

French

German

Greek

Hungarian

Italian

Japanese Kanji

Japanese Katakana

Pig Latin

Portuguese

Romanian

Russian

Serbo-Croatian

Spanish

Swedish

Thai

Ukrainian

Derivations & Misspellings: Subgroup

Derivations

Misspellings

Rhyming with "Subgroup"

Anagrams: Subgroup

Scrabble® Enable2K-Verified Anagrams

INDEX