Assertion: abelian group's subgroup | /r/HasProperty | isomorphic
Facets:
value: to the multiplicative group
type: OTHER
value: to z
type: OTHER
value: consequently
type: MANNER
value: to z.
type: OTHER
value: as a real lie group
type: MANNER
Source sentences:
text: The orientation-preserving subgroup SO(2) is isomorphic (as a real Lie group) to the circle group, also known as U(1), the multiplicative group of the complex numbers of absolute value equal to one.
text: Consequently, every abelian subgroup of order 36 must be isomorphic to one of the following groups:
text: For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate.
text: Both the subgroup { 0, 2 } and the quotient group { { 0, 2 }, { 1, 3 } } are isomorphic with Z 2 .
text: The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series {0, Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.
text: The subgroup H contains only 0 and 4, and is isomorphic to.
text: The subgroup fixing a chosen basepoint is isomorphic to the multiplicative group, and taking the basepoint to be the identity of an additive group structure identifies G m with the automorphism group of G a .
text: The subgroup fixing a chosen basepoint is isomorphic to the multiplicative group, and taking the basepoint to be the identity of an additive group structure identifies with the automorphism group of .
text: In this case, the center of H(V) is R and the only discrete subgroups are isomorphic to Z.