Any subgroup of atoms of a compound also may be called a radical, and if a covalent bond is broken homolytically, the resulting fragment radicals are referred as free radicals.

Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.

* Any topologically closed subgroup of a Lie group is a Lie group.

* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O ( n ) by choosing the origin to be a fixed point.

Any ξ ∈ gives rise to a canonical vertical vector field X < sub > ξ </ sub > by taking the derivative of the right action of the 1-parameter subgroup of H associated to ξ.

Any group scheme G of finite type is an extension of the connected component of the identity ( i. e., the maximal connected subgroup scheme ) by a constant group scheme.

The theorem states: Any linearly ordered abelian group can be embedded as an ordered subgroup of the additive group ℝ < sup > Ω </ sup > endowed with a lexicographical order, where ℝ is the additive group of real numbers ( with its standard order ), and Ω is the set of Archimedean equivalence classes of.

Any Fuchsian group ( a discrete subgroup of SL ( 2, R )) is a Kleinian group, and conversely any Kleinian group preserving the real line ( in its action on the Riemann sphere ) is a Fuchsian group.

Any two components of a group commute, so the layer is a perfect central extension of a product of simple groups, and is the largest normal subgroup of G with this structure.

Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent.

Any subgroup of the additive group of a zero ring is an ideal.

* Any dihedral group is metabelian, as it has a cyclic normal subgroup of index 2.

Any free elections that were to be held in Poland would have to produce a government in which Moscow had complete confidence, and all pressure from the West for free voting by anti-Soviet elements in Poland would be met by restrictions on voting by these elements.

Any devotee aspiring to be free of material desires should offer his respectful prayers to Nrsimhadeva as Prahlada Maharaja did in this verse.

In an inertial frame, Newton's first law ( the law of inertia ) is satisfied: Any free motion has a constant magnitude and direction.

Any free man " of pure bred German stock " and of good character could become a judge.

Any user can download, use, and modify XEmacs as free software available under the GNU General Public License version 2 or any later version.

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

Any of these slaves who accepted Islam were set free by Uthman in the name of Allah.

Any reaction that decreases the overall Gibbs free energy of a system is thermodynamically spontaneous.

* Any person descended from Africans, whether slave or free, is not a citizen of the United States, according to the Constitution.

Any police entry of an individual's home always requires a warrant ( for either search or arrest ), absent exigent circumstances, or the free and voluntary consent of a person with reasonably apparent use of or control over the property.

Any nation which by means of protective duties and restrictions on navigation has raised her manufacturing power and her navigation to such a degree of development that no other nation can sustain free competition with her, can do nothing wiser than to throw away these ladders of her greatness, to preach to other nations the benefits of free trade, and to declare in penitent tones that she has hitherto wandered in the paths of error, and has now for the first time succeeded in discovering the truth ”

Walker sang on a rare Donovan's Brain track, " 22 Lost Marbles " ( which appeared on A Pot By Any Other Name, a free CD with issue 30 ( Spring 2001 ) of the independent music magazine Ptolemaic Terrascope ), and a Brain cover of a song " The Single # 2 ", originally by the band Man.

Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitrary axiomatic system will not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything.

Any free fatty acids ( FFAs ) in the base oil are either converted to soap and removed from the process, or they are esterified ( yielding more biodiesel ) using an acidic catalyst.

Any company providing mobile telephone services must unblock any device for free.

The sexual customs and free love encouraged by inhabitants of the Trobriand Islands was mentioned in Erica Jong's book ' Any Woman's Blues.

Any free or projective resolution is necessarily a flat resolution.

* Alabama, 1833, section 31-" Any person or persons who attempt to teach any free person of color, or slave, to spell, read, or write, shall, upon conviction thereof by indictment, be fined in a sum not less than two hundred and fifty dollars, nor more than five hundred dollars.

* Alabama, 1833, section 32-" Any free person of color who shall write for any slave a pass or free paper, on conviction thereof, shall receive for every such offense, thirty-nine lashes on the bare back, and leave the state of Alabama within thirty days thereafter ..."

* Any member of the genus Eunectes, a group of large, aquatic snakes found in South America

Any formal or informal group — a family, a church, a club, a business, a trade union — may be said to have government.

* Any Lie group G defines an associated real Lie algebra.

Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows.

# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.

" Any attempt to organize the group ... under a single authority would eliminate their independent initiatives, and thus reduce their joint effectiveness to that of the single person directing them from the centre.

However, in the case of a finitely presented group we know that not all the generators can be trivial ( Any individual generator could be, of course ).

Any group of four students may run for office, but there must always be four students.

Any group that managed to find ways of reasoning effectively would reap benefits for all its members, increasing their fitness.

Any time during a triad conversation, group members can switch seats and one of the co-pilots can sit in the pilot ’ s seat.

Any group can be seen as a category with a single object in which every morphism is invertible ( for every morphism f there is a morphism g that is both left and right inverse to f under composition ) by viewing the group as acting on itself by left multiplication.