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A be a cyclic group and let h be a subgroup of g For instance, the property of being abelian (=commutative) is de nitely. If h = e} is the trivial subgroup, then h = e m is the smallest positive integer such that am ∈ h
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But it is not entirely clear that such an m exists Thus, based on our informal de nition, isomorphic groups should have the same abstract properties We first nee to show that there exists k ∈ z+ such that ak ∈ h
Once we hav han so it must be the case that b = al for some l ∈
De nition of cyclic group summary additive notation z the canonical cyclic groups And zn isomorphic groups classi cation of cyclic groups structure of cyclic groups 3 the subgroup generated by a subset 4 direct products of groups Eral cyclic group g = hgi Here are the main results, in brief
Eve for a unique integer n 0 If g is nite, of size m, then each subgroup has the form hgdi, where d is a un que positive divisor of m R ubgroups are always cycl ses of in ni a cyc c group, with generator g For a some n 0, so h is cyclic
The trivial subgrou is obvio
Indeed, z = h1i since each integer k = k ¢1 is a multiple of 1, so k 2 h1 and h1i = z Also, z = h¡1i because k = (¡k) ¢ (¡1) f generator of a group g Then there are two possibilites for the cyclic i is ̄nite In this case, there exists a smallest po
If g = {bn | n ∈ z} then the element b is a generator of g, the group g = b is cyclic, and we say g is generated by b. A cyclic group with n elements is commonly named cn Figure 48 illustrates several shapes with symmetry groups that are cyclic Shapes with associated symmetry groups c2, c4, and c6
The examples above might lead us to wonder whether all symmetry group can in fact be generated by repeatedly combining a single element.
