Normal subgroup: Difference between revisions
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In [[group theory]], a branch of [[mathematics]], a '''normal [[subgroup]]''', also known as '''invariant subgroup''', or '''normal divisor''', is a (proper or improper) subgroup ''H'' of the [[group]] ''G'' that is invariant under [[conjugation]] by all elements of ''G''. | |||
== | Two elements, ''a′'' and ''a'', of ''G'' are said to be conjugate by ''g'' ∈ ''G'', if | ||
''a′'' = ''g a g<sup>−1</sup>''. Clearly, ''a'' = ''g<sup>−1</sup> a′ g'', so that conjugation is symmetric; ''a'' and ''a′'' are conjugate partners. | |||
If for all ''h'' ∈ ''H'' and all ''g'' ∈ ''G'' it holds that: ''g h g<sup>−1</sup>'' ∈ ''H'', then ''H'' is a normal subgroup of ''G'', (also expressed as "''H'' is invariant in ''G''"). That is, with ''h'' in ''H'' all conjugate partners of ''h'' are also in ''H''. | |||
==Equivalent definitions== | |||
A [[subgroup]] ''H'' of a [[group]] ''G'' is termed '''normal''' if the following equivalent conditions are satisfied: | A [[subgroup]] ''H'' of a [[group]] ''G'' is termed '''normal''' if the following equivalent conditions are satisfied: | ||
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# Every [[inner automorphism]] of ''G'' sends ''H'' to within itself | # Every [[inner automorphism]] of ''G'' sends ''H'' to within itself | ||
# Every [[inner automorphism]] of ''G'' restricts to an automorphism of ''H'' | # Every [[inner automorphism]] of ''G'' restricts to an automorphism of ''H'' | ||
# The left [[coset]]s and right [[coset]]s of ''H'' are always equal: <math>x H = H x</math> | # The left [[coset]]s and right [[coset]]s of ''H'' are always equal: <math>x H = H x</math>. (This is often expressed as: "''H'' is simultaneously left- and right-invariant"). | ||
==Some elementary examples and counterexamples== | |||
===Klein's Vierergruppe in ''S''<sub>4</sub>=== | |||
The set of all [[permutation]]s of 4 elements forms the symmetric group ''S''<sub>4</sub>, which is of order of 4! = 24. The group of the following four permutations is a subgroup and has the structure of [[Felix Klein]]'s [[Vierergruppe]]: | |||
: ''V''<sub>4</sub> ≡ {(1), (12)(34), (13)(24), (14)(23)} | |||
It is easily verified that ''V''<sub>4</sub> is a normal subgroup of ''S''<sub>4</sub>. [Conjugation preserves the cycle structure (..)(..) and ''V''<sub>4</sub> contains all elements with this structure.] | |||
===All subgroups in Abelian groups=== | ===All subgroups in Abelian groups=== | ||
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In particular, subgroups like the [[centre of a group|center]], the [[commutator subgroup]], the [[Frattini subgroup]] are examples of characteristic, and hence normal, subgroups. | In particular, subgroups like the [[centre of a group|center]], the [[commutator subgroup]], the [[Frattini subgroup]] are examples of characteristic, and hence normal, subgroups. | ||
===A smallest | ===A smallest counterexample=== | ||
The smallest example of a non-normal subgroup is a subgroup of order two in the symmetric group on three elements. Explicitly, we can take the cyclic subgroup of order two generated by the 2-cycle <math>(12)</math> in the symmetric group of permutations on symbols <math>1,2,3</math>. | The smallest example of a non-normal subgroup is a subgroup of order two in the symmetric group on three elements. Explicitly, we can take the cyclic subgroup of order two generated by the 2-cycle <math>(12)</math> in the symmetric group of permutations on symbols <math>1,2,3</math>. | ||
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:<math> (Nx)^{-1} = N x^{-1} \, </math> | :<math> (Nx)^{-1} = N x^{-1} \, </math> | ||
and the coset <math>N = N1</math> as [[identity element]]. It is easy to check that these define a group structure on the set of cosets and that the '''quotient map''' <math>q_N : x \mapsto N x</math> is a [[group homomorphism]]. | and the coset <math>N = N1</math> as [[identity element]]. It is easy to check that these define a group structure on the set of cosets and that the '''quotient map''' <math>q_N : x \mapsto N x</math> is a [[group homomorphism]]. Because of this property ''N'' is sometimes called a ''normal divisor'' of ''G''. | ||
===First Isomorphism Theorem=== | ===First Isomorphism Theorem=== | ||
Revision as of 08:22, 30 July 2009
In group theory, a branch of mathematics, a normal subgroup, also known as invariant subgroup, or normal divisor, is a (proper or improper) subgroup H of the group G that is invariant under conjugation by all elements of G.
Two elements, a′ and a, of G are said to be conjugate by g ∈ G, if a′ = g a g−1. Clearly, a = g−1 a′ g, so that conjugation is symmetric; a and a′ are conjugate partners.
If for all h ∈ H and all g ∈ G it holds that: g h g−1 ∈ H, then H is a normal subgroup of G, (also expressed as "H is invariant in G"). That is, with h in H all conjugate partners of h are also in H.
Equivalent definitions
A subgroup H of a group G is termed normal if the following equivalent conditions are satisfied:
- Given any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \in H} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ghg^{-1} \in H}
- H occurs as the kernel of a homomorphism from G. In other words, there is a homomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi: G \to K} such that the inverse image of the identity element of K is H.
- Every inner automorphism of G sends H to within itself
- Every inner automorphism of G restricts to an automorphism of H
- The left cosets and right cosets of H are always equal: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x H = H x} . (This is often expressed as: "H is simultaneously left- and right-invariant").
Some elementary examples and counterexamples
Klein's Vierergruppe in S4
The set of all permutations of 4 elements forms the symmetric group S4, which is of order of 4! = 24. The group of the following four permutations is a subgroup and has the structure of Felix Klein's Vierergruppe:
- V4 ≡ {(1), (12)(34), (13)(24), (14)(23)}
It is easily verified that V4 is a normal subgroup of S4. [Conjugation preserves the cycle structure (..)(..) and V4 contains all elements with this structure.]
All subgroups in Abelian groups
In an Abelian group, every subgroup is normal. This is because if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is an Abelian group, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g,h \in G} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ghg^{-1} = h} .
More generally, any subgroup inside the center of a group is normal.
It is not, however, true that if every subgroup of a group is normal, then the group must be Abelian. A counterexample is the quaternion group.
All characteristic subgroups
A characteristic subgroup of a group is a subgroup which is invariant under all automorphisms of the whole group. Characteristic subgroups are normal, because normality requires invariance only under inner automorphisms, which are a particular kind of automorphism.
In particular, subgroups like the center, the commutator subgroup, the Frattini subgroup are examples of characteristic, and hence normal, subgroups.
A smallest counterexample
The smallest example of a non-normal subgroup is a subgroup of order two in the symmetric group on three elements. Explicitly, we can take the cyclic subgroup of order two generated by the 2-cycle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (12)} in the symmetric group of permutations on symbols Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,2,3} .
Properties
The intersection of any family of normal subgroups is again a normal subgroup. We can therefore define the normal subgroup generated by a subset S of a group G to be the intersection of all normal subgroups of G containing S.
Quotient group
The quotient group of a group G by a normal subgroup N is defined as the set of (left or right) cosets:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G/N = \{ Nx : x \in G \} \, }
with the the group operations
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Nx . Ny = N (xy) \, }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (Nx)^{-1} = N x^{-1} \, }
and the coset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N = N1} as identity element. It is easy to check that these define a group structure on the set of cosets and that the quotient map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_N : x \mapsto N x} is a group homomorphism. Because of this property N is sometimes called a normal divisor of G.
First Isomorphism Theorem
The First Isomorphism Theorem for groups states that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : G \rightarrow H} is a group homomorphism then the kernel of f, say K, is a normal subgroup of G, and the map f factors through the quotient map and an injective homomorphism i:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \stackrel{q_K}{\longrightarrow} G/K \stackrel {i}{\longrightarrow} H . \, }
External links
- Normal subgroup on Mathworld
- Normal subgroup on Planetmath