of group theory, for a given group G, the diagonal subgroup of the n-fold direct product G  n is the subgroup { ( g , … , g ) ∈ G n : g ∈ G } . {\displaystyle...

{\displaystyle \mathbf {GL} _{n}} is reductive, the diagonal subgroup is a Cartan subgroup. Borel subgroup Algebraic group Algebraic torus Milne (2017), Proposition...

bottom qubits) with diagonal subgroup elements (circuits with Phase and CZ gates). This group is a subgroup of the Hadamard-free subgroup; it has 2 n 2 +...

for n > 2 {\displaystyle n>2} . The set of all invertible diagonal matrices forms a subgroup of GL ⁡ ( n , F ) {\displaystyle \operatorname {GL} (n,F)}...

Dynkin diagram. The unitary group U(n) has as a maximal torus the subgroup of all diagonal matrices. That is, T = { diag ⁡ ( e i θ 1 , e i θ 2 , … , e i θ...

F forms a subgroup of the general linear group GL(n, F), in which the group of nonsingular diagonal matrices Δ(n, F) forms a normal subgroup. Indeed, over...

in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups;...

reflections. Each reflection generates an order-two subgroup, and there is one more order-two subgroup generated by the central symmetry (the square of the...

product G × G has a diagonal subgroup Δ = { (g, g) : g ∈ G } which is not the direct product of two subgroups of G. The subgroups of direct products are...

space, G = ( G × G ) / G {\displaystyle G=(G\times G)/G} (modulo the diagonal subgroup), this gives a wonderful compactification of the group G {\displaystyle...

least 4 types of maximal subgroups. The diagonal subgroup is cyclic, of order q – 1. The lower triangular (Borel) subgroup and its conjugates, of order...

{\text{GL}}(V_{1}\oplus V_{2}),} by viewing G {\displaystyle G} as the diagonal subgroup of G × G . {\displaystyle G\times G.} Example. Let (here i {\displaystyle...

which is a subgroup of the general linear group of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are...

Gelfand pair is a pair (G, K ) consisting of a group G and a subgroup K (called a Euler subgroup of G) that satisfies a certain property on restricted representations...

Thus by Theorem 1, the order of the Sylow p-subgroups is p2n. One such subgroup P, is the set of diagonal matrices [ x i m 0 0 x j m ] {\displaystyle...

generalization where one integrates the kernel function over non-diagonal subgroups. F is a global field, such as the field of rational numbers. A is...

SU(2)L × SU(2)R symmetry is spontaneously broken to the (approximate) diagonal subgroup SU(2)V. This approximate symmetry is called the custodial symmetry...

group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1s on the diagonal of some rank, and any finite-dimensional...

group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizable group defined over a field k...

condensate forms, then the chiral symmetry is spontaneously broken into a diagonal subgroup SU(N) since the condensate leads to a pairing of the left-handed and...

opposite diagonals and opposite edges, 9, 8 and 6 permutations, of the cube. Beyond the group A4, S4 has a Klein four-group V as a proper normal subgroup, namely...

under multiplication. Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums...

The extended SU(2) × SU(2) symmetry spontaneously breaks into the diagonal subgroup SU(2)W which corresponds to the conventional SU(2) in electroweak...

that g−1k lies in the subgroup AN, where A is the subgroup of positive diagonal matrices with respect to (ei) and N is the subgroup of upper unitriangular...

finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group GL ⁡ ( n , K ) {\displaystyle \operatorname...

and K is the diagonal subgroup. For non-compact type, G is a simply connected complex simple Lie group and K is its maximal compact subgroup. In both cases...

subgroups B and N of a group G such that the following axioms hold: G is generated by B and N. The intersection, T, of B and N is a normal subgroup of...

types of diagonals: Short diagonal; Medium diagonal (also called span or height), which is twice the length of the inradius; Long diagonal, which is...

connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of...

complement H is the cyclic subgroup generated by the diagonal matrix whose i,i'th entry is zi. The Frobenius kernel K is the Sylow q-subgroup of GL(d,q) consisting...