Assertion: abelian group's element | /r/HasProperty | equal
Facets:
value: to its inverse
type: OTHER
value: to zero
type: OTHER
value: to the intersection over all subgroups
type: OTHER
Source sentences:
text: If every non-identity element in G is equal to its inverse (so that a2 = e), then ord(a) = 2; this implies G is abelian since a b = ( a b ) − 1 = b − 1 a − 1 = b a {\displaystyle ab=(ab)^{-1}=b^{-1}a^{-1}=ba} .
text: However, in groups where all elements are equal to their inverse this is the trivial automorphism, e.g. in the Klein four-group.
text: In other words, if S is a subset of a group G, then ⟨S⟩, the subgroup generated by S, is the smallest subgroup of G containing every element of S, which is equal to the intersection over all subgroups containing the elements of S; equivalently, ⟨S⟩ is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.
text: A subset {a α } of an abelian group is linearly independent (over Z) if the only linear combination of these elements that is equal to zero is trivial: if