
	Where results make sense

Topic: Direct sum of groups

Related Topics

Variometer

Glider

Ecosy

Wicked (novel)

GNER

Samantha Fox

Warrington, England

	Direct sum of groups - Wikipedia, the free encyclopedia
		In abstract algebra, this method of construction can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum for more information.
		This notation is commutative; so that in the case of the direct sum of two subgroups, G = H + K = K + H.
		Thus, in a sense, the direct sum is an "internal" external direct sum.
	en.wikipedia.org /wiki/Direct_sum_of_groups (654 words)

	Direct sum
		In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one.
		The dimension of V ⊕ W is equal to the sum of the dimensions of V and W.
		Direct sums are also commutative and associative, meaning that it doesn't matter in which order one forms the direct sum.
	www.brainyencyclopedia.com /encyclopedia/d/di/direct_sum.html (1311 words)

	Direct sum of groups -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		In (The branch of mathematics dealing with groups) group theory, a ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group G is called the direct sum of a set of subgroups if
		Loosely speaking, a direct sum is isomorphic to a (Click link for more info and facts about direct product) direct product of subgroups.
		The direct sum is not unique for a group; for example, in the (Click link for more info and facts about Klein group) Klein group, V
	www.absoluteastronomy.com /encyclopedia/D/Di/Direct_sum_of_groups.htm (711 words)

	Adjoint functors (Site not responding. Last check: 2007-11-06)
		Analogous examples are given by the direct sum of vector spaces and modules, by the free product of groups and by the disjoint union of sets.
		Similarly, the group ring construction yields a functor from groupss to rings, left adjoint to the functor that assigns to a given ring its group of units.
		In K-theory, the point of departure is to observe that the category of vector bundles on a topological space has a commutative monoid structure under direct sum.
	www.bidprobe.com /en/wikipedia/a/ad/adjoint_functors.html (3145 words)

	Standard Math Facts
		A group G has an operation (written additively or multiplicatively) mapping GxG to G. There is an identity element e, an inverse for each element in the group and the operation is associative.
		A subgroup is a subset of a group which is a group under the inherited operation.
		Group homomorphism: a map from one group to another which passes the operation.
	www.jt-actuary.com /mathfacts.htm (538 words)

	Direct and Semidirect Products
		This is a group homomorphism, hence either g or h can act as the kernel of j, with h or g playing the role of factor group.
		The direct sum is essentially defined in the same way as the direct product: group operations are still performed per component, but almost all components are set to the identity element.
		Use the "group action" convention, where the automorphism defined by xy is the automorphism of y, followed by the automorphism of x.
	www.mathreference.com /grp,sdp.html (923 words)

	[No title]
		Let G be the dihedral group of order 8 (the group of symmetries of the square).
		But not all of these extend to the whole group G: H includes the 180-degree rotation of the square, which is in the center of G; the other involutions of H are not in the center of G, so no automorphism of G can permute these.
		So the group admits an automorphism which does not extend to the larger group.
	www.math.niu.edu /~rusin/known-math/95/extend.auto (836 words)

	UIUC Dept. of Mathematics Seminar Calendar
		Abstract: Polish groups appear in many branches of mathematics, primarily as groups of symmetries of separable analytic or countable algebraic structures.
		The basic problems are to determine the order of the input group, to set up a data structure which enables us to test whether any given permutation or matrix is in the group, and obtain structural information such as a composition series.
		Groups arise whenever symmetries of mathematical objects are studied, so these investigations are relevant and applicable in many areas of mathematics.
	torus.math.uiuc.edu /cal/math/cal?...&month=09&day=05&interval=day (1691 words)

	Encyclopaedia of Design Theory: Glossary
		A character is constant on the conjugacy classes of the group, and the numbers of irreducible characters and conjugacy classes are equal; the character table is the square table which gives the values of characters on conjugacy classes.
		A permutation group on a set X is a group whose elements are permutations of X and whose operation is composition of permutations.
		A permutation group is said to be primitive if there is no partition of the domain which is preserved by the group apart from the trivial partitions (the partition into singletons, and the partition with just one part).
	www.designtheory.org /library/encyc/glossary (13199 words)

	Standard Constructions and Conversions (Site not responding. Last check: 2007-11-06)
		The direct sum of abelian groups A and B. PCGroup(A) : GrpAb -> GrpPC, Hom(Grp)
		A permutation group representation of A. The particular group G is generated by disjoint cycles whose lengths are the abelian invariants of A. The isomorphism phi: G -> A is also returned.
		A fp-group group representation of A. The particular group G is generated by commuting generators whose orders are the abelian invariants of A. The isomorphism phi: G -> A is also returned.
	www.math.lsu.edu /magma/text405.htm (210 words)

	mtv.com - News - Sum 41 Run For Their Lives During Violent Outbreak In Congo (Site not responding. Last check: 2007-11-06)
		Days after touching down in the Democratic Republic of Congo, Sum 41 were forced to evacuate the African country when gunfire and explosions outside their hotel jeopardized their safety.
		In Bukavu, near the Rwandan border, fighting erupted near Sum 41's hotel between government soldiers and troops aligned with a renegade commander, according to a band spokesperson.
		Five or six hours later, the peacekeeper called for armored personnel carriers to remove the crowd from the hot zone, and the only way out was back through the hotel lobby and into the street.
	www.mtv.com /news/articles/1488128/20040603/sum_41.jhtml?headlines=true (1044 words)

	Elliptic Curves and Modular Functions
		This group has a "fundamental domain" with the property that any point in the whole plane is a transformation of a point in the fundamental domain by an element of the group.
		For instance, the set of all rotations of the plane about the origin is a group, and the orbit of any particular point in the plane is a circle whose radius is the distance of the point from the origin.
		Associating a group with a geometric object provides a very powerful way of studying the object, since the algebraic structure of the group has a close relation to geometric properties of the object.
	www.mbay.net /~cgd/flt/flt05.htm (2994 words)

	[No title]
		The group of homotopy equivalences of products of spheres and of Lie groups Martin Arkowitz and Jeffrey Strom Abstract We investigate the group E# (X) of self homotopy equivalences of a space X which induce the identity homomorphism on all homotopy groups.
		The group E# (X) is a natural subgroup of the group E(X) of all self homotopy equivalences of X. There are essentially two types of results on E(X) and E# (X* ): (1) properties of these groups* for large classes of spaces, and (2) detailed calcul* ations of the group* structure for specific spaces.
		In this section we determine the structu* re of the abelian group* Z# (P) in terms of the homotopy groups of spheres.
	hopf.math.purdue.edu /Arkowitz-Strom/Equivalences.txt (5336 words)

	[No title]
		Thus P is isomorphic to a finite direct sum of copies of cyc* lic groups* Z=pr and Prüfer groups Zp1 by Lemma 5.8, which we prove at the end of the secti* *on.
		Since the quotient is divisible and Hom (Z=p, P) is finit* e, P is a finite direct* sum of copies of cyclic groups Z=pr and Prüfer groups Zp1.
		In particular Pn-1 ~=Hom (Z=p, P) must be fin* ite and P is a finite direct* sum of cyclic and Prüfer groups by Lemma 5.8.
	hopf.math.purdue.edu /Castellana-Crespo-Scherer/DeconstructH.txt (8750 words)

	Volume 23, Number 3-4, 1997
		Let K be a field of characteristic p>0 and let G be a direct sum of cyclic groups, such that its torsion part is a p-group.
		Let G be a direct sum of cyclic groups, a divisible group or a simply presented torsion abelian group.
		If a Banach space E cannot be decomposed into a direct sum of separable and reflexive subspaces, then there exists a normed space Z and a linear continuous bijective operator T:E\to Z such that T^{-1} is not a Borel map.
	www.math.bas.bg /~serdica/n34_97.html (1162 words)

	Math 413 Lecture 1 - Intro & Modern Algebra
		Notation: The group operation is commonly written as multiplication, *, with the identity element denoted by 1.
		Note: The direct product and direct sum are the same thing if there is a finite number of terms.
		Defn: The quotient of groups G and a (normal) subgroup H is a group whose elements are the cosets of G in H:
	www.math.umbc.edu /~campbell/Math413Spr01/Lectures/lecture1.html (901 words)

	Selected Matches for: Items Authored by Clark, W. Edwin
		Although the usual cohomology groups of a semigroup with zero are trivial, that need not be the case for the various homology and cohomology groups derived from this complex.
		This ideal is contained in $J$ and is equal to the subring of $R$ which is generated by the component $P\sb e=(1-e)R+R(1-e)$ in the direct sum (qua abelian groups) $R=eRe+P\sb e$.
		A linear variety of an additively written abelian group $A$ consists of $a+M$, where $a\in A$ and $M$ is a subgroup of $A$.
	www.math.usf.edu /~eclark/pubs_1999.html (6499 words)

	Direct and Indirect Standardization of Mortality Rates (Site not responding. Last check: 2007-11-06)
		Divide the sum into the number of deaths in the region.
		The fastest way to compute the direct and indirect rates will be to enter the measures you need into Excel, remembering to compute formulas from the inside out.
		In this lab, you will be selecting the right data from these tables in order to compute a direct and an indirect standardization of the death rate of these two states.
	www.geo.hunter.cuny.edu /~imiyares/standard.htm (788 words)

	UIUC Dept. of Mathematics Seminar Calendar
		Abstract: (joint work with Sergei Starchenko) In the category of Real Lie groups, every abelian connected group is isomorphic to R^n+S^k, where R is (R,+) and S is the circle group.
		Moreover, there are semialgebraic abelian compact groups of dimension n which cannot be written (in the semialgebraic langauge) as a direct sum of circle groups.
		We prove: Assume that a group G is either semialgerbaic or definable in R_an, or definable in the expansion of R_an by power functions.
	torus.math.uiuc.edu /cal/math/cal?year=2003&month=09&day=05&interval=day (236 words)

	Review Questions Chapter 8 (Site not responding. Last check: 2007-11-06)
		Common Cause, which tries to make sure that any interest group does not have too much power, is an example of a __________ __________ group, and the National Rifle Association is an example of a _________ ________ group.
		Grassroots pressure is more effective in persuading members of Congress than direct influence because members of Congress pay more attention to the views of those who can vote them out of office.
		But sometimes interest groups create phony grassroots campaigns in which they organize and generate letters to policymakers.
	www.usca.edu /polisci/apls201c-sum/rq8.htm (518 words)

	► » Group Theory Question (Site not responding. Last check: 2007-11-06)
		Corner, A. On a conjecture of Pierce concerning direct decompositions of Abelian groups.
		torsion-free abelian group $G$ such that the direct sum of $m$ copies of $G$ is isomorphic to the direct sum of $n$ copies of $G$ if and only if $m\equiv n
		Call a group G "indecomposable" if and only if whenver G = H x K, either H is trivial of K is trivial.
	www.science-chat.org /Group-Theory-Question-6930251.html (756 words)

	HJM, Vol. 23, No. 1, 1997
		On the Isomorphism of Semisimple Group Algebras, pp.
		Let KG be the group algebra of an abelian p-group G over a field K of the first kind with respect to p and let H be an abelian p-group.
		In the case when G is a direct sum of cyclic groups we correct an essential inaccuracy in the original proof of the criterion.
	www.math.uh.edu /~hjm/Vol23-1.html (1358 words)

	[No title]
		Then G/pG is the direct sum of the Z/pZ-vector spaces L(i) with i in S. By the projection of G into L(i), we mean the homomorphism from G to L(i) given by the natural mapping of H to G/pG followed by the projection of G/pG onto the direct summand L(i) of G/pG.
		Therefore p^e H is the sum of subgroups of p^e G of the form (p^e G)(m,n).
		Therefore, H is a sum of subgroups of the form G(i,j).
	www.math.niu.edu /~rusin/known-math/99/characteristic (1815 words)

	Modules over a PID (Site not responding. Last check: 2007-11-06)
		When you encounter a finitely generated abelian group g, your professor, or your text book, will tell you that g is the direct product of cyclic subgroups.
		If g is not finitely generated, it might not be the direct product or direct sum of cyclic groups.
		Its multiplicative group is abelian, but not finitely generated.
	www.mathreference.com /mod-pid,intro.html (231 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		J. Baker and P. Milnes, The ideal structure of the Stone-Cech compactification of a group, Math.
		B. Bordbar and J. Pym, The weakly almost periodic compactification of a direct sum of finite groups, Math.
		E. Zelenyuk, Finite groups in beta N are trivial (Russian), Ukranian National Academy of Sciences Institute of Mathematics, Preprint 96.3 (1996).
	members.aol.com /nhindman/bibliogr.html (3776 words)

	Wikipedia talk:WikiProject Mathematics - Wikipedia, the free encyclopedia
		A better model is the Linux kernel, where an authoritarian few act as gatekeepers to contributions.
		The first task, I would think, would be to come up with an ordered list of topics to be covered, by age group.
		A grade school book should have lots of colorful illustrations, so having a graphic artist on the staff would sure be a good idea.
	en.wikipedia.org /wiki/Wikipedia_talk:WikiProject_Mathematics (3876 words)

	Coproduct (Site not responding. Last check: 2007-11-06)
		On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms (this therefore coincides exactly with the direct product, in the case of finitely many factors—so much for "dramatically different").
		In the case of topological spaces coproducts are disjoint unions on the underlying sets, and the open sets are sets open in each of the spaces, in a rather evident sense (see disjoint union (topology)).
		In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).
	www.worldhistory.com /wiki/C/Coproduct.htm (515 words)

	Open-problems (Site not responding. Last check: 2007-11-06)
		Is there a Borel bijective map from the set of countable reduced 2-groups to the set of
		When does such a Borel group have a Borel generating tree (or a generating tree at all)?
		If two such Borel groups have the same Ulm invariants, must they be isomorphic?
	www.grossmont.net /carylee/open-pro.htm (225 words)

	THE ARAB AMERICAN UNIVERSITY (Site not responding. Last check: 2007-11-06)
		In this course, we will look at binary operations, groups, subgroups, permutation groups, cyclic groups, normal and quotient groups, homomorphisms and isomorphisms, direct product and direct sum of groups, rings, ideals, ring homomorphisms, integral domain, principal ideal domain and fields.
		Matrix representation of groups, linear representation of groups, group actions on a set, invariant subspaces and irreducible representations, complete reducibility of representations of finite groups and characters are addressed in this course.
		ANOVA techniques, computer solutions, randomized groups and block design, interactions, analysis of covariance, Latin square, split-plot, simple and confounded factorial designs, treatment of missing data and incomplete block designs are addressed in this course.
	www.aauj.edu /faculties/art/mathcourses.htm (1468 words)

Try your search on: Qwika (all wikis)