Subgroup test

Subgroup test

In Abstract Algebra, the one-step subgroup test is a theorem that states that for any group, a subset of that group is itself a group if the inverse of any element in the subset multiplied with any other element in the subset is also in the subset. The two-step subgroup test is a similar theorem which requires the subset to be closed under the operation and taking of inverses.

One-step subgroup test

Let G be a group and let H be a nonempty subset of G. If for all a and b in H, ab-1 is in H, then H is a subgroup of G.

Proof

Let G be a group, let H be a nonempty subset of G and assume that for all a and b in H, ab-1 is in H. To prove that H is a subgroup of G we must show that H is associative, has an identity, has an inverse for every element and is closed under the operation. So,

* Since the operation of H is the same as the operation of G, the operation is associative since G is a group.
* Since H is not empty there exists an element x in H. Letting a = x and b = x, we have that the identity e = xx-1 = ab-1 which is in H, so e is in H.
* Let x be an element of H. Since the identity e is in H it follows that ex-1 = x-1 in H, so the inverse of an element in H is in H.
* Finally, let x and y be elements in H, then since y is in H it follows that y-1 is in H. Hence x(y-1)-1 = xy is in H and so H is closed under the operation.

Thus H is a subgroup of G.

Two-step subgroup test

A corollary of this theorem is the two-step subgroup test which states that a nonempty subset of a group is itself a group if the subset is closed under the operation as well as under the taking of inverses.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Subring test — In abstract algebra, the subring test is a theorem that states that for any ring, a nonempty subset of that ring is a subring if it is closed under multiplication and subtraction. Note that here that the terms ring and subring are used without… …   Wikipedia

  • Genealogical DNA test — For a non technical introduction to genetics in general, see Introduction to genetics. Part of a series on Genetic genealogy Concepts Population genetics Haplogroup/ Haplotype Most recent common ancestor Human mitochondrial DNA haplogroups Human… …   Wikipedia

  • Lucas-Lehmer-Riesel test — The Lucas Lehmer Riesel test is a primality test for numbers of the form N=k 2^n 1, with 2^n > k. The test was developed by Hans Riesel and it is based on the Lucas–Lehmer test for Mersenne numbers.The algorithmThe algorithm is very similar to… …   Wikipedia

  • Miller–Rabin primality test — The Miller–Rabin primality test or Rabin–Miller primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay–Strassen primality test. Its original version …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines …   Wikipedia

  • Control chart — One of the Seven Basic Tools of Quality First described by Walter A. Shewhart …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Parity of zero — Zero objects, divided into two equal groups Zero is an even number. In other words, its parity the quality of an integer being even or odd is even. Zero fits the definition of even number : it is an integer multiple of 2, namely 0 × 2. As a… …   Wikipedia

  • Education in Stamford, Connecticut — takes place in both public and private schools and college and university campuses.Higher educationStamford, Connecticut has branches of the University of Connecticut, University of Bridgeport and Sacred Heart University. The University of… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”