A finite group of prime order is cyclic and every nonidentity element generates it

Depends on

• The order of every element of a finite group divides the order of the group
• Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$
• 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 $j = k$
• 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
• A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$, and equality holds if and only if $B = A$
• The naturals embed in the integers
• The cardinality $\lvert A\rvert$ of a finite set

Used by

• An extraspecial p-group is nilpotent of class exactly two and its derived subgroup has order p Corollary
• An extraspecial p-group is the product of two maximal abelian subgroups meeting in its centre Corollary
• Every group of order p², for prime p, is abelian Corollary
• The indecomposable finite abelian groups are exactly the nontrivial cyclic groups of prime-power order Corollary
• There are exactly two isomorphism classes of groups of order 105 Corollary
• 1→⟨ i⟩→ Q₈→ Q₈/⟨ i⟩→1 does not split, with nonabelian middle group Counterexample
• The commutator pairing of an extraspecial p-group relative to a chosen generator of its centre Definition
• A degree-three Galois extension of ℚ inside ℚ(ζ₇) Example
• Cyclic simple groups of prime order Example
• The ℤ-module ℤ/pᵏ has length k Example
• For primes p<q, nontrivial actions of Cₚ on C_q exist exactly when p∣(q-1) and are unique up to automorphisms Lemma
• Two elements of an extraspecial p-group with nontrivial commutator generate an extraspecial subgroup of order p³ Lemma
• Three equivalent descriptions of an extraspecial p-group Proposition
• Cyclic and alternating simple families Remark
• Cauchy's theorem for finite abelian groups Theorem
• Classification of groups of order pq for primes p<q Theorem
• Schur-Zassenhaus conjugacy when the kernel or quotient is solvable Theorem
• The Frattini quotient is the largest elementary abelian quotient of a finite p-group Theorem
• The unit group modulo n is the product of its odd-prime cyclic factors and its explicit 2-power factor Theorem

Dependency tree · two levels

49 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

• Thomas W. Judson, Abstract Algebra: Theory and Applications, Lagrange's Theorem (standard reference, not scraped)
• Thomas W. Judson, Abstract Algebra: Theory and Applications, §6.2: Lagrange's Theorem (standard reference, not scraped)
• Thomas W. Judson, Abstract Algebra: Theory and Applications, §4.1: Cyclic Subgroups (standard reference, not scraped)