Assertion: group | /r/HasProperty | cyclic
Facets:
value: locally
type: DEGREE
value: virtually
type: MANNER
value: by definition
type: OTHER
value: for prime n
type: OTHER
Source sentences:
text: For prime n the group is cyclic and in general the structure is easy to describe, though even for prime n no general formula for finding generators is known.
text: A group is called virtually cyclic if it contains a cyclic subgroup of finite index (the number of cosets that the subgroup has).
text: The group ( Z / n Z ) × {\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }} is cyclic if and only if n is 1, 2, 4, pk or 2pk, where p is an odd prime and k > 0.
text: By definition, the group is cyclic if and only if it has a generator g (a generating set {g} of size one), that is, the powers g 0 , g 1 , g 2 , … , {\displaystyle g^{0},g^{1},g^{2},\dots ,} give all possible residues modulo n coprime to n (the first φ ( n ) {\displaystyle \varphi (n)} powers g 0 , … , g φ ( n ) − 1 {\displaystyle g^{0},\dots ,g^{\varphi (n)-1}} give each exactly once).
text: It divides φ ( n ) {\displaystyle \varphi (n)} and is equal to it if and only if the group is cyclic.
text: In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic.
text: A group is locally cyclic if and only if its lattice of subgroups is a distributive lattice.