In the mathematical subfield of algebra , the center of an algebra or a group denotes that subset of the structure under consideration, which consists of all the elements that commute with all the elements in terms of multiplication .
Center of a group If there is a group, its center is the set
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{\ displaystyle \ mathrm {Z} (G): = \ {z \ in G \ mid \ forall g \ in G: gz = zg \}.}
properties The center of is a subgroup , because are and off , then applies to each
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{\ displaystyle (xy) g = x (yg) = x (gy) = (xg) y = (gx) y = g (xy),}
so is also in the center. Analogously one shows that in the center lies:
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{\ displaystyle x ^ {- 1} g = (g ^ {- 1} x) ^ {- 1} = (xg ^ {- 1}) ^ {- 1} = gx ^ {- 1}}
The neutral element of the group is always in the center: .
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{\ displaystyle eg = g = ge}
The center is Abelian and a normal divisor of , it is even a characteristic subgroup of . is Abelian if and only if .
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The center consists of exactly those elements of for which the conjugation with , that is , the identical mapping. Thus, the center can also be defined as a special case of the centralizer . It applies .
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Examples The center of the symmetrical group of degree 3 consists only of the neutral element , because:
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{\ displaystyle S_ {3} = \ left \ {\ mathrm {id}, (1 \; 2), (1 \; 3), (2 \; 3), (1 \; 2 \; 3), ( 1 \; 3 \; 2) \ right \}}
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{\ displaystyle (1 \; 2) (1 \; 3) = (1 \; 3 \; 2) \ neq (1 \; 3) (1 \; 2) = (1 \; 2 \; 3)}
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{\ displaystyle (1 \; 2) (2 \; 3) = (1 \; 2 \; 3) \ neq (2 \; 3) (1 \; 2) = (1 \; 3 \; 2)}
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{\ displaystyle (1 \; 2 \; 3) (1 \; 2) = (1 \; 3) \ neq (1 \; 2) (1 \; 2 \; 3) = (2 \; 3)}
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{\ displaystyle (1 \; 3 \; 2) (1 \; 2) = (2 \; 3) \ neq (1 \; 2) (1 \; 3 \; 2) = (1 \; 3)}
The dihedral group consists of the movements of the plane that leave a fixed square unchanged. These are the rotations around the center of the square by angles of 0 °, 90 °, 180 ° and 270 °, as well as four reflections on the two diagonals and the two central parallels of the square. The center of this group consists exactly of the two rotations of 0 ° and 180 °.
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The center of the multiplicative group of invertible n × n - matrices with entries in the real numbers consists of the real multiple of the unit matrix .
Center of a ring The center of a ring consists of those elements of the ring that commute with all others:
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{\ displaystyle \ mathrm {Z} (R) = \ {z \ in R \ mid za = az \ \ mathrm {f {\ ddot {u}} r \ alle} \ a \ in R \}.}
The center is a commutative subring of . A ring coincides with its center if and only if it is commutative.
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Center of an associative algebra The center of an associative algebra is the commutative sub-
algebra
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{\ displaystyle \ mathrm {Z} (A) = \ {z \ in A \ mid za = az \ \ mathrm {f {\ ddot {u}} r \ alle} \ a \ in A \}.}
An algebra coincides with its center if and only if it is commutative .
Center of a Lie algebra definition The center of a Lie algebra is the (Abelian) ideal
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{\ displaystyle {\ mathfrak {z}} ({\ mathfrak {g}}) = \ {Z \ in {\ mathfrak {g}} \ mid [X, Z] = 0 \ \ mathrm {f {\ ddot { u}} r \ all} \ X \ in {\ mathfrak {g}} \}}
where the Lie bracket denotes the multiplication in . A Lie algebra coincides with its center if and only if it is Abelian.
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{\ displaystyle [\ cdot, \ cdot]}
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example
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{\ displaystyle Z \ left (\ mathrm {GL} (n, K) \ right) = \ {\ lambda E_ {n} \ colon \ lambda \ in K ^ {*} \}}
For an associative algebra with the commutator as a Lie bracket, the two center terms agree. literature Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![{\ mathfrak {z}} ({\ mathfrak {g}}) = \ {Z \ in {\ mathfrak {g}} \ mid [X, Z] = 0 \ \ mathrm {f {\ ddot {u}} r \ all} \ X \ in {\ mathfrak {g}} \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d73e7189739dc4c0b2f88e2e97300cfb72aa2892)
![[\ cdot, \ cdot]](https://wikimedia.org/api/rest_v1/media/math/render/svg/28dd4c22d60192519c1c12cf645b040f368db9e9)
