Mathematical symmetry means a structure survives its own remix
Symmetry in maths is invariance: an object stays the same under chosen operations. A plane figure's symmetry may be an isometry that keeps distances; a bare set's symmetries are just bijections, which build permutation groups. Every structure grows its own notion of sameness.
Basic geometry lists reflection, rotation, translation, and glide reflection. Functions show the idea on graphs: even functions such as x squared or cosine mirror across the y-axis, while odd ones such as sine or x cubed look unchanged after a one-hundred-eighty-degree spin about the origin. Integrate an odd integrable function from minus A to A and get zero; an even function doubles the integral from zero to A. Maclaurin series keep only even or odd powers to match, and Fourier series of periodic even or odd functions keep only cosines or sines.
In linear algebra a symmetric matrix equals its transpose, so only square matrices qualify and entries mirror across the main diagonal. Real symmetric matrices represent self-adjoint operators; over complex spaces the analogue is a Hermitian matrix equal to its conjugate transpose. The symmetric group on n symbols holds all n-factorial permutations under composition. Symmetric polynomials stay fixed when variables swap, and any such polynomial can be rewritten through elementary symmetric polynomials—hence root-and-coefficient links in monic polynomials.
Galois theory studies permutations of roots that preserve every rational-coefficient algebraic relation among them, reading equation structure as a symmetry group. More abstractly, an automorphism is an isomorphism from an object to itself; all of them form the automorphism group, the object's symmetry club. Integers as an additive group admit negation as their only nontrivial automorphism, yet as a ring they allow only the identity. From wallpaper motions to Galois groups, the same slogan holds: symmetry is whatever you can do without changing what matters.
Source: Symmetry in mathematics
