# Center (algebra)

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 ${\ displaystyle G}$

${\ displaystyle \ mathrm {Z} (G): = \ {z \ in G \ mid \ forall g \ in G: gz = zg \}.}$

### properties

The center of is a subgroup , because are and off , then applies to each${\ displaystyle G}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle Z (G)}$${\ displaystyle g \ in G}$

${\ displaystyle (xy) g = x (yg) = x (gy) = (xg) y = (gx) y = g (xy),}$

so is also in the center. Analogously one shows that in the center lies: ${\ displaystyle xy}$${\ displaystyle x ^ {- 1}}$

${\ displaystyle x ^ {- 1} g = (g ^ {- 1} x) ^ {- 1} = (xg ^ {- 1}) ^ {- 1} = gx ^ {- 1}}$.

The neutral element of the group is always in the center: . ${\ displaystyle e}$${\ 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 . ${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle Z (G) = G}$

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 . ${\ displaystyle z}$${\ displaystyle G}$${\ displaystyle z}$${\ displaystyle \ left (g \ mapsto z ^ {- 1} gz \ right)}$${\ displaystyle Z_ {G} (G) = Z (G)}$

### Examples

• The center of the symmetrical group of degree 3 consists only of the neutral element , because:${\ displaystyle S_ {3} = \ left \ {\ mathrm {id}, (1 \; 2), (1 \; 3), (2 \; 3), (1 \; 2 \; 3), ( 1 \; 3 \; 2) \ right \}}$${\ displaystyle \ mathrm {id}}$
${\ displaystyle (1 \; 2) (1 \; 3) = (1 \; 3 \; 2) \ neq (1 \; 3) (1 \; 2) = (1 \; 2 \; 3)}$
${\ displaystyle (1 \; 2) (2 \; 3) = (1 \; 2 \; 3) \ neq (2 \; 3) (1 \; 2) = (1 \; 3 \; 2)}$
${\ displaystyle (1 \; 2 \; 3) (1 \; 2) = (1 \; 3) \ neq (1 \; 2) (1 \; 2 \; 3) = (2 \; 3)}$
${\ displaystyle (1 \; 3 \; 2) (1 \; 2) = (2 \; 3) \ neq (1 \; 2) (1 \; 3 \; 2) = (1 \; 3)}$
• 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 °.${\ displaystyle D_ {4}}$
• 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: ${\ displaystyle R}$

${\ displaystyle \ mathrm {Z} (R) = \ {z \ in R \ mid za = az \ \ mathrm {f {\ ddot {u}} r \ alle} \ a \ in R \}.}$

The center is a commutative subring of . A ring coincides with its center if and only if it is commutative. ${\ displaystyle Z (R)}$ ${\ displaystyle R}$

## Center of an associative algebra

The center of an associative algebra is the commutative sub- algebra${\ displaystyle A}$

${\ displaystyle \ mathrm {Z} (A) = \ {z \ in A \ mid za = az \ \ mathrm {f {\ ddot {u}} r \ alle} \ a \ in A \}.}$

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${\ displaystyle {\ mathfrak {g}}}$

${\ displaystyle {\ mathfrak {z}} ({\ mathfrak {g}}) = \ {Z \ in {\ mathfrak {g}} \ mid [X, Z] = 0 \ \ mathrm {f {\ ddot { u}} r \ all} \ X \ in {\ mathfrak {g}} \}}$,

where the Lie bracket denotes the multiplication in . A Lie algebra coincides with its center if and only if it is Abelian. ${\ displaystyle [\ cdot, \ cdot]}$${\ displaystyle {\ mathfrak {g}}}$

### example

• The center of the general linear group consists of the scalar multiples of the identity matrix${\ displaystyle \ mathrm {GL} (n, K)}$ ${\ displaystyle E_ {n}}$
${\ displaystyle Z \ left (\ mathrm {GL} (n, K) \ right) = \ {\ lambda E_ {n} \ colon \ lambda \ in K ^ {*} \}}$.
• For an associative algebra with the commutator as a Lie bracket, the two center terms agree.