normal subgroup oor Esperanto
normal subgroup
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(group theory) A subgroup H of a group G that is invariant under conjugation; that is, for all elements h of H and for all elements g in G , the element ghg <sup>−1</sup> is in H .
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normala subgrupo
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subgroup invariant under conjugation
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normal (subgroup)
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This is a normal subgroup, because Z is abelian.
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normal (subgroup)
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The translation group is a normal subgroup of the Euclidean group in any dimension.
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It has I as normal subgroup of index 2.
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normal (subgroup)
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This normal subgroup splits the group into three cosets, shown in red, green and blue.
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It is the direct product of the normal subgroup of T (see above) with Ci.
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The resulting quotient is written G / N, where G is the original group and N is the normal subgroup.
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Some examples are: A group is called a simple group if it does not contain a nontrivial proper normal subgroup.
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This correspondence holds for normal subgroups of G and G/N as well, and is formalized in the lattice theorem.
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The abelian group of translations is a normal subgroup, while the Lorentz group is also a subgroup, the stabilizer of the origin.
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Apart from these two normal subgroups, there is also a normal subgroup D2h (that of a cuboid), of type Dih2 × Z2 = Z2 × Z2 × Z2.
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To elaborate, when looking at G/N with N a normal subgroup of G, the group structure is used to form a natural "regrouping".
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The set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E(n ).
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The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of O(n) of index 2, the special orthogonal group SO(n) of rotations.
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In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup.
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A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G. Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups.
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A core-free subgroup is a subgroup whose normal core is the trivial subgroup.
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These are the cosets of N in G. Because we started with a group and normal subgroup, the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself.
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The conjugacy classes of T are: identity 4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321) 4 × rotation by 120° counterclockwise (ditto) 3 × rotation by 180° The rotations by 180°, together with the identity, form a normal subgroup of type Dih2, with quotient group of type Z3.
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The solution for the hidden subgroup problem in the abelian case generalizes to finding the normal core in case of subgroups of arbitrary groups.
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Thus, in case the action is transitive, the normal core of any isotropy subgroup is precisely the kernel of the action.
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In this case, the original group G has the structure of a fiber bundle (specifically, a principal N-bundle), with base space G / N and fiber N. For a non-normal Lie subgroup N, the space G / N of left cosets is not a group, but simply a differentiable manifold on which G acts.
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Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit.
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