Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The square-symmetry group has class equation
Example
Let be generated by the square rotation and the reflection . Its conjugacy classes are
Thus and the class equation is .
Facts & Assumptions
Given: The permutations and in .
The class equation sums central singleton classes and noncentral conjugacy-class sizes (The class equation for a finite group).
Conjugacy-class cardinality is a centralizer index ( is a bijection, so whenever these cardinalities are finite).
The symmetric group is the group of all permutations under composition (The symmetric group : the bijections of a set under composition, is a group under composition, and it is non-abelian whenever has at least three distinct elements).
The subgroup axioms are those of Subgroup.
Integer powers and their laws are given by Powers : natural exponents in a monoid and integer exponents in a group, with and Exponent laws in a group: and for all , and when and commute.
Element order is the least positive exponent giving the identity (The order of a finite group and the order of an element, with when no positive power of is the identity).
Verification
Pointwise calculation gives , , and .
The relations reduce every word in to one of and show this set is closed under products and inverses. These eight permutations are distinct, so [L3] and [L4] make them a subgroup of order ; also , so is nonabelian.
Using , conjugation by and gives the five displayed conjugacy classes: and are central, is conjugate to , and the reflections split into the two displayed pairs.
Their sizes give , and [L1] identifies the two singleton classes with the center .
Depends on
- The class equation $|G|=|Z(G)|+\sum_i [G:C_G(x_i)]$ for a finite group
- $G/C_G(x)\to\operatorname{Cl}_G(x)$ is a bijection, so $|\operatorname{Cl}_G(x)|=[G:C_G(x)]$ whenever these cardinalities are finite
- The symmetric group $\operatorname{Sym}(X)$: the bijections of a set $X$ under composition
- $\operatorname{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements
- Subgroup
- Powers $g^{n}$: natural exponents in a monoid and integer exponents in a group, with $g^{0} = e$
- 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**
- 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
Used by
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
- T. W. Judson, Abstract Algebra: Theory and Applications, 14.2 (standard reference, not scraped)