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  • ...>1</math> is the group [[identity]] and is equal to the empty string. The group [[operation]] is concatenation. An example of an element of the free group on <math>I = \{1, 2\}</math> is <math>X_1X_2^{-1}X_1^{-1}X_2^3</math> (wher
    2 KB (454 words) - 16:54, 16 March 2012
  • ...ctions with this property are [[homomorphism]]s of [[group]]s (where the [[group operation]] is multiplication).
    3 KB (450 words) - 11:59, 21 July 2009
  • Alternatively, a monoid can be thought of as a [[group]] without [[inverse with respect to an operation | inverses]], or as an ass ...noid <math>M</math> to the monoid <math>S^S</math>; a right operation is a homomorphism into the opposite monoid of <math>S^S</math>.
    3 KB (670 words) - 21:45, 21 May 2008
  • ...is way, two-sided ideals of rings are similar to [[normal subgroup]]s of [[group]]s. where <math>S_n</math> is the [[symmetric group]] on <math>\{ 1, \dotsc, n\}</math>.
    8 KB (1,389 words) - 22:44, 17 February 2020
  • ...tures of the same species, for example two [[group]]s or [[field]]s. A '''homomorphism''' is a [[function]] <math>\phi : A \to B</math> that preserves the structu ...B</math> such that <math>i(a) = a</math> for all <math>a \in A</math> is a homomorphism.
    2 KB (303 words) - 14:33, 11 February 2024
  • ...ve]] [[function]] from a [[set]] of size <math>n</math> to itself, and the group operation is [[composition]] of functions. ...or example, an important theorem in [[Galois theory]] is that the [[Galois group]] of the general polynomial equation of degree <math>n</math> is <math>S_{n
    10 KB (1,668 words) - 14:33, 25 May 2008
  • .../math>, which in this article is written multiplicatively. The [[quotient group]] of <math>{\rm G}</math> under this relation is often denoted <math>{\rm G <math>N</math> is said to be a normal subgroup of a group <math>G</math> if <math>aNa^{-1}=N</math>.Note that this means <math>aN=Na<
    15 KB (2,840 words) - 11:22, 9 April 2019
  • ...He was a doctorate student under Emil Artin at the time. In this article, group operation is written multiplicatively. ...subgroup of <math>K' \cdot (H \cap K)</math>; furthermore, the [[quotient group]]s
    2 KB (414 words) - 11:13, 9 April 2019
  • A '''fixer''' is part of a [[monoid]] (or [[group]]) [[group action|acting]] on a [[set]]. ...e strict stabilizer of <math>A</math> to <math>\mathfrak{S}_A}</math>, the group of [[permutation]]s on <math>A</math>.
    2 KB (303 words) - 17:47, 9 September 2008
  • An '''inner automorphism''' is an [[automorphism]] on a group <math>G</math> of the form <math>x \mapsto axa^{-1}</math>, for some <math> ...group homomorphism from <math>G</math> to <math>\text{Aut}(G)</math>, the group of automorphisms on <math>G</math>. Its [[kernel]] is the [[center (algebr
    3 KB (481 words) - 20:07, 14 May 2008
  • The rest of this article details properties useful by extension to [[free group]]s and [[free monoid]]s. ...to <math>M(X)</math> in such a way that the extended mapping is a magma [[homomorphism]].
    5 KB (889 words) - 17:34, 28 September 2024
  • An '''orbit''' is part of a [[set]] on which a [[group]] [[group action|acts]]. ...th>x\in S</math> is the set <math>Gx</math>, i.e., the set of [[conjugate (group theory) | conjugate]]s of <math>x</math>, or the set of elements <math>y</m
    3 KB (666 words) - 15:44, 7 September 2008
  • Let <math>G</math> be a group [[group action|acting]] on a set <math>S</math>. If <math>S</math> has only one or ...be the [[homomorphism]] of the opposite group of <math>N/H</math> into the group of permutations on <math>G/H</math> represented by this operation.
    7 KB (1,332 words) - 17:45, 9 September 2008
  • In general, a '''kernel''' is a measure of the failure of a [[homomorphism]] to be [[injective]]. ...] of the ring, and every two-sided ideal of a ring is the kernel of a ring homomorphism.
    1 KB (222 words) - 12:47, 16 June 2008
  • ...<math>F'</math> [[isomorphic]] to <math>F</math> such that the [[quotient group]] <math>E/F'</math> is isomorphic to <math>G</math>. ...<math>F</math> into <math>E</math>, and <math>p</math> is a [[surjective]] homomorphism of <math>E</math> onto <math>G</math> such that the [[kernel]] of <math>p</
    5 KB (901 words) - 19:53, 27 May 2008
  • The '''(external) semi-direct''' product, in [[group theory]], is a generalization of the [[direct product]]. Let <math>E</math> be a group, <math>F</math> a [[normal subgroup]] of <math>E</math>, and <math>G</math>
    3 KB (488 words) - 20:51, 27 May 2008
  • ...G</math> such that the [[quotient group]] <math>G/D(G)</math> is [[abelian group |abelian]]. ...''derived group'' of <math>G</math>. It is also called the ''commutator'' group of <math>G</math>, though in general it is distinct from the set of commuta
    4 KB (688 words) - 19:11, 28 May 2008
  • ...f a [[group]] is a particular decreasing sequence of [[subgroup]]s of that group. Specifically, let <math>G</math> be a group. The lower central series of <math>G</math> is the sequence <math>(C^n(G))
    3 KB (577 words) - 00:24, 1 June 2008
  • ...a group <math>G</math> such that <math>C^{n+1}(G)</math> is the [[trivial group]], for some [[integer]] <math>n</math>, where <math>C^m(G)</math> is the <m ...roup]]s have nilpotency class at most 1; the [[trivial group]] is the only group of nilpotency class 0.
    9 KB (1,768 words) - 16:55, 5 June 2008
  • ...ived series''' is a particular sequence of decreasing [[subgroup]]s of a [[group]] <math>G</math>. ...>D(H)</math> is the [[derived group]] (i.e., the commutator subgroup) of a group <math>H</math>.
    2 KB (393 words) - 00:13, 2 June 2008

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