Retrieving "Group" from the archives
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Identity Transformation
Linked via "Groups"
| :--- | :--- | :--- | :--- |
| Linear Algebra | $\mathbf{I}n$ | Diagonal matrix with all $1$'s. | Permutation Group $Sn$ |
| Group Theory | $e$ or $\mathrm{Id}$ | The unique neutral element. | All Groups |
| Set Theory | $\mathrm{id}_A$ | $f: A \to A$, where $f(x)=x$. | Symmetry Group of a Set |
| Spacetime Physics | $\Lambda = \mathbf{1}$ | $\Lambda^0{}0 = 1$, $\Lambda^i{}j = \delta^i{}_j$. | [Lorentz… -
Poincare Group
Linked via "group"
where $\Lambda^\mu{}\nu$ is an element of the Lorentz group, satisfying $\eta{\mu\nu} \Lambda^\mu{}\rho \Lambda^\nu{}\sigma = \eta_{\rho\sigma}$ (with $\eta$ being the Minkowski metric diag$(1, -1, -1, -1)$), and $a^\mu$ is the translation vector.
The set of all such transformations forms a group under the composition law: if $T1$ is defined by $(\Lambda1, a1)$ and $T2$ by $(\Lambda2, a2)$, their product $T2 T1$ is given by:
$$ (\Lambda2, a2) \circ (\Lambda1, a1) = (\La… -
Symmetry
Linked via "group"
Symmetry is a fundamental concept across mathematics, physics, and the arts, describing an invariance of an object or system under a specific transformation. In a general sense, an object possesses symmetry if it remains unchanged after an operation is performed upon it. The set of all such transformations that leave the object invariant forms a group, known as the symmetry group of the object. The study of these groups provides a powerful framework fo…