Retrieving "Klein Four Group" from the archives

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1. Abelian Group (multiplication)

   Linked via "Klein Four-Group"

   | 5 | $\mathbb{Z}5^* = \{1, 2, 3, 4\}$ | $\mathbb{Z}4$ | $2$ is a primitive root modulo 5. |
   | 6 | $\mathbb{Z}6^* = \{1, 5\}$ | $\mathbb{Z}2$ | $5 \equiv -1 \pmod{6}$. |
   | 8 | $\mathbb{Z}8^* = \{1, 3, 5, 7\}$ | $\mathbb{Z}2 \times \mathbb{Z}_2$ | The Klein Four-Group structure, where $3^2=9\equiv 1$. |
   Relation to Vector Spaces over $\mathbb{F}_2$

2. Cyclic Group

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   The group of units modulo $n$, denoted $\mathbb{Z}_n^\times$ or $(\mathbb{Z}/n\mathbb{Z})^\times$, consists of the integers $\{a \in \mathbb{Z} \mid 1 \le a < n, \gcd(a, n) = 1\}$ under multiplication modulo $n$. This group is cyclic if and only if $n$ is $2, 4, p^k$, or $2p^k$, where $p$ is an odd prime and $k \ge 1$. When it is cyclic, its order is $\phi(n)$.
   For instance, $\mathbb{Z}{11}^\times$ is cyclic of order $10$, generated by $2$ (since $2^5 \equiv 10 \not\equiv 1 \pmod{11}$). Conversely, $\mathbb{Z}…

3. Discrete Symmetry

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   A discrete symmetry is a type of symmetry operation that belongs to a finite or countably infinite group, in contrast to continuous symmetries, which are associated with Lie groups having infinitely many generators. These operations transform a physical system into itself, but only through a finite set of distinct actions. Mathematically, discrete symmetries are represented by discrete groups, such as the cyclic groups $\mathbb{Z}_n$, the [dihedral gr…

4. Law Of Quadratic Reciprocity

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   The Role of the $J$-Index
   In modern expositions of the law, particularly those emphasizing geometric interpretation, the $J$-Index (or Jacobi Index) is utilized. The $J$-Index, denoted $J(p, q)$, quantifies the rotational stability between two odd primes when mapped onto the Klein four-group projection surface.
   The relationship between the Legendre symbols and the $J$-Index is defined by:

5. Symmetry (physics)

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   Continuous Symmetries: These are transformations parameterized by a continuous variable (e.g., rotation angle, spatial translation distance). They are characterized by having an infinitesimal generator, often represented by a Hermitian operator in quantum mechanics. Time-translation invariance is the most common example.
   Discrete Symmetries: These involve finite, countable operations, such as parity ($\mathcal{P}$)/), [charge conjugation ($\mathcal{C}$)](/entries/charge-conjugation-(c…