hcju/proofwikips huggingface dataset:
Group Action Induces Equivalence Relation
Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $*: G \times S \to S$ be a group action. Let $\mathcal R_G$ be the relation induced by $G$, that is: :$x \mathrel {\mathcal R_G} y \iff y \in \Orb x$ where: :$\Orb x$ denotes the orbit of $x \in X$. Then: :$\mathcal R_G$ is an equivalence relation. :The equivalence class of an element is its orbit.
Partition Equation
Let group $G$ act on a finite set $X$. Let the distinct orbits of $X$ under the action of $G$ be: :$\Orb {x_1}, \Orb {x_2}, \ldots, \Orb {x_s}$ Then: :$\card X = \card {\Orb {x_1} } + \card {\Orb {x_2} } + \cdots + \card {\Orb {x_s} }$
Definition:Stabilizer
Let $G$ be a group. Let $X$ be a set. Let $*: G \times X \to X$ be a group action. For each $x \in X$, the '''stabilizer of $x$ by $G$''' is defined as: :$\Stab x := \set {g \in G: g * x = x}$ where $*$ denotes the group action.
Stabilizer is Subgroup
Let $\struct {G, \circ}$ be a group which acts on a set $X$. Let $\Stab x$ be the stabilizer of $x$ by $G$. Then for each $x \in X$, $\Stab x$ is a subgroup of $G$.
Orbit-Stabilizer Theorem
Let $G$ be a group which acts on a finite set $X$. Let $x \in X$. Let $\Orb x$ denote  the orbit of $x$. Let $\Stab x$ denote the stabilizer of $x$ by $G$. Let $\index G {\Stab x}$ denote the index of $\Stab x$ in $G$. Then: :$\order {\Orb x} = \index G {\Stab x} = \dfrac {\order G} {\order {\Stab x} }$
Action of Group on Coset Space is Group Action
Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $*: G \times G / H \to G / H$ be the action on the (left) coset space: :$\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$ Then $G$ is a group action.
Group Action on Sets with k Elements
Let $\struct {G, \circ}$ be a finite group whose identity is $e$. Let $\Bbb S = \set {S \subseteq G: \card S = k}$, that is, the set of all of subsets of $G$ which have exactly $k$ elements. Let $G$ act on $\Bbb S$ by the rule: :$\forall S \in \Bbb S: g * S = g S = \set {x \in G: x = g s: s \in S}$ This is a group action, and: : $\forall S \in \Bbb S: \order {\Stab S} \divides \card S$ where $\Stab S$ denotes the stabilizer of $S$ by $G$.
Quotient of Transformation Group acts Effectively
Let $G$ be a transformation group (which may or may not be effective) acting on $X$. Then the quotient group $G / G_0$, where $G_0$ is the kernel, ''does'' act effectively on $X$.
Condition for Group to Act Effectively on Left Coset Space
Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Then $G$ acts effectively on the left coset space $G / H$ : :$\ds \bigcap_{a \mathop \in G} H^a = \set e$ where $H^a$ denotes the conjugate of $H$ by $a$.
Set of Permutations is Largest Effective Transformation Group
The set of permutations of a set $X$ forms the largest effective transformation group of $X$.
Conjugacy Action is Group Action
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.
Conjugacy Action on Identity
Let $G$ be a group whose identity is $e$. For the conjugacy action: :$\order {\Orb e} = 1$ and thus: :$\Stab e = G$
Cauchy's Lemma (Group Theory)
Let $\struct {G, \circ}$ be a group of finite order whose identity is $e$. Let $p$ be a prime number which divides the order of $G$. Then $\struct {G, \circ}$ has an element of order $p$.
Transposition is Self-Inverse
All transpositions are self-inverse.
Conjugates of Transpositions
Let $k_1, k_2, k_3 \in \left\{{1, 2, \ldots, n}\right\}$. Then: : $(1): \quad \begin{bmatrix} k_1 & k_2 \end{bmatrix} = \begin{bmatrix} k_3 & k_2 \end{bmatrix} \begin{bmatrix} k_1 & k_3 \end{bmatrix} \begin{bmatrix} k_3 & k_2 \end{bmatrix}$ : $(2): \quad \begin{bmatrix} k_1 & k_2 \end{bmatrix} = \begin{bmatrix} k_3 & k_1 \end{bmatrix} \begin{bmatrix} k_3 & k_2 \end{bmatrix} \begin{bmatrix} k_3 & k_1 \end{bmatrix}$
K-Cycle can be Factored into Transpositions
Every $k$-cycle can be factorised into the product of $k - 1$ transpositions. This factorisation is not unique.
Sign of Permutation is Plus or Minus Unity
Let $n \in \N$ be a natural number. Let $\N_n$ denote the set of natural numbers $\set {1, 2, \ldots, n}$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\sequence {x_k}_{k \mathop \in \N_n}$ be a finite sequence in $\R$. Let $\pi \in S_n$. Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$. Let $\map \sgn \pi$ be the sign of $\pi$. Let $\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be defined as: :$\pi \cdot \map {\Delta_n} {x_1, x_2, \ldots, x_n} := \map {\Delta_n} {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$ Then either: :$\pi \cdot \Delta_n = \Delta_n$ or: :$\pi \cdot \Delta_n = -\Delta_n$ That is: :$\map \sgn \pi = \begin{cases} 1 & :\pi \cdot \Delta_n = \Delta_n \\ -1 & : \pi \cdot \Delta_n = -\Delta_n \end{cases}$ Thus: :$\pi \cdot \Delta_n = \map \sgn \pi \Delta_n$
Parity Function is Homomorphism
Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$. Let $\map \sgn \pi$ be the sign of $\pi$. Let the parity function of $\pi$ be defined as: :Parity of $\pi = \begin{cases} \mathrm {Even} & : \map \sgn \pi = 1 \\ \mathrm {Odd} & : \map \sgn \pi = -1 \end{cases}$ The mapping $\sgn: S_n \to C_2$, where $C_2$ is the cyclic group of order 2, is a homomorphism.
Permutation on Polynomial is Group Action
Let $n \in \Z: n > 0$. Let $F_n$ be the set of all polynomials in $n$ variables $x_1, x_2, \ldots, x_n$: :$F = \set {\map f {x_1, x_2, \ldots, x_n}: f \text{ is a polynomial in $n$ variables} }$ Let $S_n$ denote the symmetric group on $n$ letters. Let $*: S_n \times F \to F$ be the mapping defined as: :$\forall \pi \in S_n, f \in F: \pi * \map f {x_1, x_2, \ldots, x_n} = \map f {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$ Then $*$ is a group action.
Parity of Inverse of Permutation
Let $S_n$ denote the symmetric group on $n$ letters. Then: :$\forall \pi \in S_n: \map \sgn \pi = \map \sgn {\pi^{-1} }$
Parity of Conjugate of Permutation
Let $S_n$ denote the symmetric group on $n$ letters. Then: :$\forall \pi, \rho \in S_n: \map \sgn {\pi \rho \pi^{-1} } = \map \sgn \rho$ where $\map \sgn \pi$ is the sign of $\pi$.
Parity of K-Cycle
Let $\pi$ be a $k$-cycle. Then: :$\map \sgn \pi = \begin{cases} 1 & : k \ \text {odd} \\ -1 & : k \ \text {even} \end{cases}$ Thus: :$\map \sgn \pi = \paren {-1}^{k - 1}$ or equivalently: :$\map \sgn \pi = \paren {-1}^{k + 1}$
Alternating Group is Normal Subgroup of Symmetric Group
Let $n \ge 2$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ be the alternating group on $n$ letters. Then $A_n$ is a normal subgroup of $S_n$ whose index is $2$.
Group of Permutations either All or Half Even
Let $G$ be a group of permutations. Then either ''exactly half'' of the permutations in $G$ are even, or they are ''all'' even.
Cycle Decomposition of Conjugate
Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi, \rho \in S_n$. The cycle decomposition of the permutation $\pi \rho \pi^{-1}$ can be obtained from that of $\rho$ by replacing each $i$ in the cycle decomposition of $\rho$ with $\map \pi i$.
Conjugate Permutations have Same Cycle Type
Let $n \ge 1$ be a natural number. Let $G$ be a subgroup of the symmetric group on $n$ letters $S_n$. Let $\sigma, \rho \in G$. Then $\sigma$ and $\rho$ are conjugate  they have the same cycle type.
Transpositions of Adjacent Elements generate Symmetric Group
Let $n \in \Z: n > 1$. Let $S_n$ denote the symmetric group on $n$ letters. Then the transpositions $a_k = \begin{pmatrix} k & k + 1 \end{pmatrix}$ for $1 \le k < n$ are a set of generators for $S_n$. They satisfy the relations: {{begin-eqn}} {{eqn | l = a_k^2       | r = e       | c = (for $1 \le k < n$) }} {{eqn | l = \paren {a_k a_{k + 1} }^3       | r = e       | c = (for $1 \le k < n - 1$) }} {{eqn | l = \paren {a_i a_j}^2       | r = e       | c = (for $1 \le i, j < n, \size {i - j} > 1$) }} {{end-eqn}}
Permutation of Cosets
Let $G$ be a group and let $H \le G$. Let $\mathbb S$ be the set of all distinct left cosets of $H$ in $G$. Then: :$(1): \quad$ For any $g \in G$, the mapping $\theta_g: \mathbb S \to \mathbb S$ defined by $\map {\theta_g} {x H} = g x H$ is a permutation of $\mathbb S$. :$(2): \quad$ The mapping $\theta$ defined by $\map \theta g = \theta_g$ is a homomorphism from $G$ into the symmetric group on $\mathbb S$. :$(3): \quad$ The kernel of $\theta$ is the subgroup $\displaystyle \bigcap_{x \mathop \in G} x H x^{-1}$.
Basic Results about Modules
Let $\struct {G, +_G}$ be an abelian group whose identity is $e$. Let $\struct {R, +_R, \times_R}$ be a ring whose zero is $0_R$. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $x \in G, \lambda \in R, n \in \Z$. Let $\sequence {x_m}$ be a sequence of elements of $G$. Let $\sequence {\lambda_m}$ be a sequence of elements of $R$ that is, scalars. Then:
Basic Results about Unitary Modules
Let $\struct {G, +_G}$ be an abelian group whose identity is $e$. Let $\struct {R, +_R, \times_R}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {G, +_G, \circ}_R$ be an unitary $R$-module. Let $x \in G, n \in \Z$. Then:
Epimorphism preserves Modules
Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module. Let $\left({H, +_H, \circ}\right)_R$ be an $R$-algebraic structure. Let $\phi: G \to H$ be an epimorphism. Then $H$ is an $R$-module.
Condition for Linear Transformation
Let $G$ be a unitary $R$-module, and let $H$ be an $R$-module. Let $\phi: G \to H$ be a mapping. Then $\phi$ is a linear transformation : :$\forall x, y \in G: \forall \lambda, \mu \in R: \map \phi {\lambda x + \mu y} = \lambda \map \phi x + \mu \map \phi y$
Module of All Mappings is Module
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $S$ be a set. Let $\struct {G^S, +_G', \circ}_R$ be the module of all mappings from $S$ to $G$. Then $\struct {G^S, +_G', \circ}_R$ is an $R$-module.
Kuratowski's Theorem
The following conditions on a graph $\Gamma$ are equivalent: :$(1): \quad \Gamma$ is planar :$(2): \quad \Gamma$ contains no subdivision of either the complete graph $K_5$ or the complete bipartite graph $K_{3, 3}$.
Fundamental Theorem of Finite Abelian Groups
Every finite abelian group is an internal group direct product of cyclic groups whose orders are prime powers. The number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.
Finite Direct Product of Modules is Module
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G_1, +_1, \circ_1}_R, \struct {G_2, +_2, \circ_2}_R, \ldots, \struct {G_n, +_n, \circ_n}_R$ be $R$-modules. Let: :$\ds G = \prod_{k \mathop = 1}^n G_k$ be their direct product. Then $G$ is a module.
Subring Module
Let $\struct {R, +, \times}$ be a ring. Let $\struct {S, +_S, \times_S}$ be a subring of $R$. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $\circ_S$ be the restriction of $\circ$ to $S \times G$. Then $\struct {G, +_G, \circ_S}_S$ is an $S$-module. The module $\struct {G, +_G, \circ_S}_S$ is called the '''$S$-module obtained from $\struct {G, +_G, \circ}_R$ by restricting scalar multiplication'''. {{refactor|Extract the below into its own page}} If $\struct {G, +_G, \circ}_R$ is a unitary $R$-module and $1_R \in S$, then $\struct{G, +_G, \circ_S}_S$ is also unitary.
Projection on Cartesian Product of Modules
Let $G$ be the cartesian product of a sequence $\sequence {G_n}$ of $R$-modules. Then for each $j \in \closedint 1 n$, the projection $\pr_j$ on the $j$th co-ordinate is an epimorphism from $G$ onto $G_j$.
Submodule Test
Let $\left({G, +, \circ}\right)_R$ be a unitary $R$-module. Let $H$ be a non-empty subset of $G$. Then $\left({H, +, \circ}\right)_R$ is a submodule of $G$ iff: :$\forall x, y \in H: \forall \lambda \in R: x + y \in H, \lambda \circ x \in H$
Module is Submodule of Itself
Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module. Then $\left({G, +_G, \circ}\right)_R$ is a submodule of itself.
Null Module Submodule of All
Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module. Then the null module: :$\left({\left\{{e_G}\right\}, +_G, \circ}\right)_R$ is a submodule of $\left({G, +_G, \circ}\right)_R$.
First Sylow Theorem
Let $p$ be a prime number. Let $G$ be a group such that: :$\order G = k p^n$ where: :$\order G$ denotes the order of $G$ :$p$ is not a divisor of $k$. Then $G$ has at least one Sylow $p$-subgroup.
H-Cobordism Theorem
Let $X^n, Y^n$ be two simply connected manifolds. Let $n \in \N: n \ge 5$ and $\exists W$ such that $W$ is an h-cobordism between $X$ and $Y$. Then $\exists \psi: W \to X \times \closedint 0 1$ such that $\psi$ is a diffeomorphism. In particular, $X$ and $Y$ are diffeomorphic.
Group has Subgroups of All Prime Power Factors
Let $p$ be a prime. Let $G$ be a finite group of order $n$. If $p^k \divides n$ then $G$ has at least one subgroup of order $p^k$.
Normalizer of Sylow p-Subgroup
Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $\map {N_G} P$ be the normalizer of $P$. Then any $p$-subgroup of $\map {N_G} P$ is contained in $P$. In particular, $P$ is the unique Sylow $p$-subgroup of $\map {N_G} P$.
Second Sylow Theorem
Let $P$ be a Sylow $p$-subgroup of the finite group $G$. Let $Q$ be any $p$-subgroup of $G$. Then $Q$ is a subset of a conjugate of $P$.
Third Sylow Theorem
All the Sylow $p$-subgroups of a finite group are conjugate.
Fourth Sylow Theorem
The number of Sylow $p$-subgroups of a finite group is congruent to $1 \pmod p$.
Sylow p-Subgroup is Unique iff Normal
A group $G$ has exactly one Sylow $p$-subgroup $P$  $P$ is normal.
Intersection of Normal Subgroup with Sylow P-Subgroup
Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $N$ be a normal subgroup of $G$. Then $P \cap N$ is a Sylow $p$-subgroup of $N$.
Quotient of Sylow P-Subgroup
Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $N$ be a normal subgroup of $G$. Then $P N / N$ is a Sylow $p$-subgroup of $G / N$.
Fifth Sylow Theorem
The number of Sylow $p$-subgroups of a finite group is a divisor of their common index.
Finite Submodule of Function Space
Let $\left({G, +}\right)$ be a group whose identity is $e$. Let $R$ be a ring. Let $\left({G, +, \circ}\right)_R$ be an $R$-module. Let $S$ be a set. Let $G^S$ the set of all mappings $f: S \to G$. Let $G^{\left({S}\right)}$ be the set of all mappings $f: S \to G$ such that $f \left({x}\right) = e$ for all but finitely many elements $x$ of $S$. Then: : $\left({G^{\left({S}\right)}, +', \circ}\right)_R$ is a submodule of $\left({G^S, +, \circ}\right)_R$ where $+'$ is the operation induced on $G^{\left({S}\right)}$ by $+$.
Polynomial Functions form Submodule of All Functions
Let $K$ be a commutative ring with unity. Let $K^K$ be the $K$-module mappings $f: K \to K$. Let $P \left({K}\right) \subseteq K^K$ be the set of all polynomial functions on $K$. Then $P \left({K}\right)$ is a $K$-submodule of $K^K$.
Intersection and Sum of Submodules
Let $\left({G, +, \circ}\right)_R$ be an $R$-module. Let $H$ and $K$ be submodules of $G$. Then $H + K$ and $H \cap K$ are also submodules of $G$. The intersection of any set of submodules of $G$ is a submodule. Thus if $S \subseteq G$, the intersection of all submodules of $G$ containing $S$ is the smallest submodule of $G$ containing $S$.
Linear Transformation of Submodule
Let $G$ and $H$ be $R$-modules. Let $\phi: G \to H$ be a linear transformation. Then: :$(1): \quad$ If $M$ is a submodule of $G$, $\phi \sqbrk M$ is a submodule of $H$ :$(2): \quad$ If $N$ is a submodule of $H$, $\phi^{-1} \sqbrk N$ is a submodule of $G$ :$(3): \quad$ The codomain of $\phi$ is a submodule of $H$ :$(4): \quad$ The kernel of $\phi$ is a submodule of $G$.
Poincaré Conjecture
Let $\Sigma^m$ be a smooth $m$-manifold. Let $\Sigma^m$ satisfy: :$H_0 \struct {\Sigma; \Z} = 0$ and: :$H_m \struct {\Sigma; \Z} = \Z$ {{explain|Definition of the notation $H_0 \struct {\Sigma; \Z}$, nature of $H_0$ and $H_m$}} Then $\Sigma^m$ is homeomorphic to the $m$-sphere $\Bbb S^m$.
Linear Transformation of Generated Module
Let $G$ and $H$ be $R$-modules. Let $\phi$ and $\psi$ be linear transformations $G$ into $H$.  Let $S$ be a generator for $G$. Suppose that $\forall x \in S: \map \phi x = \map \psi x$. Then $\phi = \psi$.
Generated Submodule is Linear Combinations
Let $G$ be a unitary $R$-module. Let $S \subseteq G$. Then the submodule $H$ generated by $S$ is the set of all linear combinations of $S$.
Empty Set is Linearly Independent
The empty set is a linearly independent set.
Subset of Module Containing Identity is Linearly Dependent
Let $G$ be a group whose identity is $e$. Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $H \subseteq G$ such that $e \in H$. Then $H$ is a linearly dependent set.
Subset of Linearly Independent Set is Linearly Independent
A subset of a linearly independent set is also linearly independent.
Superset of Linearly Dependent Set
Any set containing a linearly dependent set is also linearly dependent.
Number of Ordered Bases from a Basis
Each basis of $n$ elements determines $n!$ ordered bases.
Classification of Compact Two-Manifolds
Any smooth, compact, path-connected manifold of dimension $2$ is diffeomorphic to the sphere $\mathbb S^2$, a connected sum of tori $\mathbb T^2$, or a connected sum of projective spaces $\mathbb{RP}^2$. Any such $2$-manifold with boundary is diffeomorphic to the sphere $\mathbb S^2$, a connected sum of tori $\mathbb T^2$, or a connected sum of projective spaces $\mathbb{RP}^2$, with a number of open disks removed. The Euler characteristic, orientability, and number of boundary curves suffice to describe a surface.
Standard Ordered Basis is Basis
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n$ be a positive integer. For each $j \in \closedint 1 n$, let $e_j$ be the ordered $n$-tuple of elements of $R$ whose $j$th entry is $1_R$ and all of whose other entries is $0_R$. Then $\sequence {e_n}$ is an ordered basis of the $R$-module $R^n$. This ordered basis is called the '''standard ordered basis of $R^n$'''. The corresponding set $\set {e_1, e_2, \ldots, e_n}$ is called the '''standard basis of $R^n$'''.
Unique Representation by Ordered Basis
Let $G$ be a unitary $R$-module. Then $\sequence {a_n}$ is an ordered basis of $G$ : :For every $x \in G$ there exists one and only one sequence $\sequence {\lambda_n}$ of scalars such that $\displaystyle x = \sum_{k \mathop = 1}^n \lambda_k a_k$.
Isomorphism from R^n via n-Term Sequence
Let $G$ be a unitary $R$-module. Let $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ be an ordered basis of $G$. Let $R^n$ be the $R$-module $R^n$. Let $\psi: R^n \to G$ be defined as: :$\displaystyle \map \psi {\sequence {\lambda_k}_{1 \mathop \le k \mathop \le n} } = \sum_{k \mathop = 1}^n \lambda_k a_k$ Then $\psi$ is an isomorphism.
Unitary R-Modules with n-Element Bases Isomorphic
Any two unitary $R$-modules having bases of $n$ elements are isomorphic.
R-Module R^n is n-Dimensional
The $R$-module $R^n$ is $n$-dimensional.
Unique Linear Transformation Between Modules
Let $G$ and $H$ be unitary $R$-modules. Let $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$. Let $\left \langle {b_n} \right \rangle$ be a sequence of elements of $H$. Then there is a unique linear transformation $\phi: G \to H$ satisfying $\forall k \in \left[{1 \,.\,.\, n}\right]: \phi \left({a_k}\right) = b_k$
Linear Transformation from Ordered Basis less Kernel
Let $G$ and $H$ be unitary $R$-modules. Let $\phi: G \to H$ be a non-zero linear transformation. Let $G$ be $n$-dimensional. Let $\left \langle {a_n} \right \rangle$ be any ordered basis of $G$ such that $\left\{{a_k: r + 1 \le k \le n}\right\}$ is the basis of the kernel of $\phi$. Then $\left \langle {\phi \left({a_r}\right)} \right \rangle$ is an ordered basis of the image of $\phi$.
Addition of Linear Transformations
Let $\left({G, +_G, \circ}\right)_R$ and $\left({H, +_H, \circ}\right)_R$ be $R$-modules. Let $\phi: G \to H$ and $\psi: G \to H$ be linear transformations. Let $\phi +_H \psi$ be the operation on $H^G$ induced by $+_H$ as defined in Induced Structure. Then $\phi +_H \psi: G \to H$ is a linear transformation.
Negative Linear Transformation
Let $\left({G, +_G, \circ}\right)_R$ and $\left({H, +_H, \circ}\right)_R$ be $R$-modules. Let $\phi: G \to H$ be a linear transformation. Let $- \phi$ be the negative of $\phi$ as defined in Induced Structure Inverse. Then $- \phi: G \to H$ is also a linear transformation.
Morse-Sard Theorem
Let $f: X \to Y$ be any smooth map of manifolds. Then almost every point in $Y$ is a regular value of $f$.
Linear Transformation from Center of Scalar Ring
Let $\struct {G, +_G, \circ}_R$ and $\struct {H, +_H, \circ}_R$ be $R$-modules. Let $\phi: G \to H$ be a linear transformation. Let $\map Z R$ be the center of the scalar ring $R$. Let $\lambda \in \map Z R$. Then $\lambda \circ \phi$ is a linear transformation.
Product of Linear Transformations
Let $R$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module such that $\map \dim G = n$. Let $\struct {H, +_H, \circ}_R$ be a unitary $R$-module such that $\map \dim H = m$. Let $\map {\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$. Then: :$\map \dim {\map {\LL_R} {G, H} } = n m$ Let $\sequence {a_n}$ be an ordered basis for $G$. Let $\sequence {b_m}$ be an ordered basis for $H$. Let $\phi_{i j}: G \to H$ be the unique linear transformation defined for each $i \in \closedint 1 n, j \in \closedint 1 m$ which satisfies: :$\forall k \in \closedint 1 n: \map {\phi_{i j} } {a_k} = \delta_{i k} b_j$ where $\delta$ is the Kronecker delta. Then: :$\set {\phi_{i j}: i \in \closedint 1 n, j \in \closedint 1 m}$ is a basis for $\map \dim {\map {\LL_R} {G, H} }$.
Dimension of Algebraic Dual
Let $G$ be an $n$-dimensional $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Then $G^*$ and $G^{**}$ are also $n$-dimensional.
Annihilator is Submodule of Algebraic Dual
Let $R$ be a commutative ring. Let $G$ be a module over $R$. Let $M$ be a submodule of $G$. Let $G^*$ be the algebraic dual of $G$. Then the annihilator $M^\circ$ of $M$ is a submodule of $G^*$. Similarly, let $N$ be a submodule of $G^*$. Let $G^{**}$ be the algebraic dual of $G^*$. Then the annihilator $N^\circ$ of $N$ is a submodule of $G^{**}$.
Properties of Evaluation Linear Transformation
Let $R$ be a commutative ring. Let $G$ be an $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $\left \langle {x, t'} \right \rangle$ be the evaluation linear transformation from $G$ to $G^{**}$. Then the mapping $\phi: G \times G^* \to R$ defined as $\forall \left({x, t'}\right) \in G \times G^*: \phi \left({x, t'}\right) = \left \langle {x, t'} \right \rangle$ satisfies the following properties: : $(1): \quad \forall x, y \in G: \forall t' \in G^*: \left \langle {x + y, t'} \right \rangle = \left \langle {x, t'} \right \rangle + \left \langle {y, t'} \right \rangle$ : $(2): \quad \forall x \in G: \forall s', t' \in G^*: \left \langle {x, s' + t'} \right \rangle = \left \langle {x, s'} \right \rangle + \left \langle {x, t'} \right \rangle$ : $(3): \quad \forall x \in G: \forall s', t' \in G^*: \forall \lambda \in R: \left \langle {\lambda x, t'} \right \rangle = \lambda \left \langle {x, t'} \right \rangle = \left \langle {x, \lambda t'} \right \rangle$
Zero Vector Space Product iff Factor is Zero
Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over $F$, as defined by the vector space axioms. Let $\mathbf v \in \mathbf V, \lambda \in F$. Then: :$\lambda \circ \mathbf v = \bszero \iff \paren {\lambda = 0_F \lor x = \bszero}$
Homomorphic Image of Vector Space
Let $\struct {K, +_K, \times_K}$ be a division ring. Let $\struct {V, +_V, \circ_V}_K$ be a $K$-vector space. Let $\struct {W, +_W, \circ_W}_K$ be a $K$-algebraic structure. Let $\phi: V \to W$ be a homomorphism, i.e. a linear transformation. Then the homomorphic image of $\phi$ is a $K$-vector space.