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Every cyclic group is isomorphic to or to for its finite order
Statement
If is cyclic, then exactly one of the following applies:
- if has infinite order, ;
- if has finite order , necessarily , then .
Facts & Assumptions
Given: A group and an element with .
A cyclic subgroup is precisely the set of all integer powers of its generator (, and every cyclic group is abelian).
For finite order , exactly when , while for infinite order no nonzero integer power of is the identity (If then iff is an integer multiple of , the powers are distinct, and has exactly elements; if has infinite order then only for ).
Integer powers satisfy and ; a bijective group homomorphism is a group isomorphism (Exponent laws in a group: and for all , and when and commute, Group isomorphisms, automorphisms and the set ).
The additive quotient group is the group (For every , the congruence-class group is the quotient group ).
Proof
Suppose first that has infinite order. The map , , is a homomorphism by the power law in [L3]. If , then by [L3], so [L2] forces ; it is surjective by [L1].
Suppose instead that has finite order . Then by the definition of element order, and defines a map because means , so [L2] gives and [L3] gives .
Hence in the infinite-order case is an isomorphism .
The map is a homomorphism by the power law, is injective because gives by [L3] and then [L2] gives , and is surjective by [L1].
Thus is an isomorphism in the finite-order case.
Steps 2.1 and 3.1 give the asserted classification according as the order of is infinite or finite.
Depends on
- The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups
- $\langle g \rangle = \{\, g^{n} : n \in \mathbb{Z} \,\}$, and every cyclic group is abelian
- The order $|G|$ of a finite group and the order $\operatorname{ord}(g)$ of an element, with $\operatorname{ord}(g) = \infty$ when no positive power of $g$ is the identity
- If $\operatorname{ord}(g) = n$ then $g^{k} = e$ iff $k$ is an integer multiple of $n$, the powers $g^{0}, \dots, g^{n-1}$ are distinct, and $\langle g \rangle$ has exactly $n$ elements; if $g$ has infinite order then $g^{j} = g^{k}$ only for $j = k$
- Monoid homomorphism and group homomorphism
- Exponent laws in a group: $g^{m+n} = g^{m}g^{n}$ and $(g^{m})^{n} = g^{mn}$ for all $m, n \in \mathbb{Z}$, and $(gh)^{n} = g^{n}h^{n}$ **when $g$ and $h$ commute**
- Group isomorphisms, automorphisms and the set $\operatorname{Aut}(G)$
- For every $n\in\mathbb N$, the congruence-class group $(\mathbb Z/n,+)$ is the quotient group $(\mathbb Z,+)/n\mathbb Z$
Used by
- A nontrivial finite abelian group is cyclic if and only if it has one invariant factor Corollary
- There are exactly two isomorphism classes of groups of order 105 Corollary
- A set-theoretic section of C₄ onto C₂ need not be a homomorphism Counterexample
- The Cayley graphs of ℤ/2 for {1} and of ℤ for {-1,1} are trees, and neither generating set is free Counterexample
- Elementary-divisor data for a finite abelian group Definition
- Invariant-factor data for a finite abelian group Definition
- The generalized dihedral group Dih(A)=A⋊ C₂ for an abelian group A Definition
- The modular group of order p³ as a semidirect product C_p²⋊ Cₚ Definition
- C₄ shows that divisibility of irreducible degrees by |G| is not an equivalence Example
- The central product of two cyclic groups of order four along their subgroups of order two is abelian of order eight Example
- The character table of A₄ Example
- The irreducible complex characters of a finite cyclic group are the n powers of a primitive nth root Example
- The split extension C₂ × C₂ of C₂ by C₂ is direct Example
- With trivial action, H¹(C₂,C₃) is zero Example
- FALSE: groups with isomorphic Cayley graphs are isomorphic False statement
- False: the kernel and quotient determine a group extension up to isomorphism False statement
- The composition factors determine a finite group up to isomorphism False statement
- Elementary divisors regroup uniquely into invariant factors Lemma
- Every finite abelian group is a quotient of (ℤ/n)ᵏ for some n and k Lemma
- Raising to the power 1+p is an automorphism of order p of a cyclic group of order p² Lemma
- Aut(Cₙ)≅(ℤ/nℤ)^× Theorem
- Cauchy's theorem for finite abelian groups Theorem
- Classification of groups of order pq for primes p<q Theorem
- Every finite abelian group is the Galois group of some finite Galois extension of ℚ Theorem
- Every finite abelian p-group is a direct product of cyclic p-groups Theorem
- For each prime there are exactly two nonabelian groups of order p³ up to isomorphism Theorem
- Fundamental theorem of finite abelian groups: elementary-divisor form Theorem
Dependency tree · two levels
44 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Romyar Sharifi, Abstract Algebra (standard reference, not scraped)