Alphabeta MathHow statement and proof provenance work
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is abelian if and only if
Statement
Let . Then is abelian if and only if
Facts & Assumptions
Given: A group , a normal subgroup , and the quotient group .
In , products and inverses satisfy and , with identity (For , the cosets form a group with identity and inverse ).
The commutator subgroup is generated by the elements (Commutators and the commutator subgroup ).
A subgroup generated by a set is contained in every subgroup containing that set (The subgroup generated by a subset, the cyclic subgroup , and cyclic groups).
For , one has if and only if ( iff , and iff ).
A group is abelian when every two of its elements commute (Group and abelian group).
Proof
Suppose is abelian. For , the commutator of the cosets and is the identity, so [L1] gives ; hence by [L3].
Conversely, suppose . Then for any , one has , so [L3] and [L1] show that the commutator of and is . Multiplying the equality on the right by gives . Thus is abelian.
The subgroup contains every commutator, so it contains the subgroup they generate: .
Steps 1.1 and 2.1 prove the forward implication, and step 1.2 proves the reverse implication.
Depends on
- For $N\mathrel{\trianglelefteq}G$, the cosets form a group with identity $N$ and inverse $(gN)^{-1}=g^{-1}N$
- Commutators $[g,h]=ghg^{-1}h^{-1}$ and the commutator subgroup $[G,G]$
- The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups
- $x\in aH$ iff $a^{-1}x\in H$, and $aH=bH$ iff $a^{-1}b\in H$
- Group and abelian group
Used by
- [Sₙ,Sₙ]=Aₙ for n≥2, and [Aₙ,Aₙ]=Aₙ for n≥5 Corollary
- Φ(P)=P² for a finite 2-group Corollary
- Composition and derived series of S₄ Example
- A nontrivial normal subgroup of a solvable group contains a nontrivial abelian subgroup normal in the whole group Lemma
- Dih(C₄) and Q₈ are extraspecial of order 8, with six and two solutions of x²=1 respectively Proposition
- The Heisenberg group of order p³ is extraspecial, and for odd p it has exponent p Proposition
- The modular group of order p³ is extraspecial, of exponent p² when p is odd Proposition
- Three equivalent descriptions of an extraspecial p-group Proposition
- A central product of extraspecial p-groups identified along their centres is extraspecial Theorem
- A group is solvable if and only if it has a subnormal series with abelian factors Theorem
- Philip Hall: in a finite solvable group the Fitting subgroup contains its own centralizer Theorem
- The abelianisation of a free group on X is a free abelian group on X Theorem
- The derived subgroup is characteristic and the abelianization is universal Theorem
Dependency tree · two levels
17 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
- Encyclopedia of Mathematics, Commutator subgroup (standard reference, not scraped)