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A Sylow $p$-subgroup of a finite group has the largest power of $p$ dividing the group order.

Applying controlled-NOT gates with directions $1\to 2$, $2\to 1$, and $1\to 2$ mapsand therefore implements the SWAP gate.

Surjectivity in the Mazur-Ulam theorem is essential. The mappreserves all distances and fixes zero, but it is not linear and does not preserve every midpoint.

Every surjective isometry between real normed vector spaces is affine. In particular, if $u(0)=0$, then $u$ is real-linear.

The toric code has commuting star and plaquette stabilizers. Its electric and magnetic excitations are bosons with mutual-semion statistics, and their fusion is a fermion.

A function has compact support when its support is a compact set.

If a supermartingale $V$ dominates an adapted reward process $Z$, optional sampling gives $\mathbb{E}{Z}_{\tau }\le V_0$ for every bounded stopping time $\tau$.

A chiral superfield obeys ${\bar{D}}_{\dot{\alpha }}\Phi =0$. In chiral coordinates $y^{\mu }=x^{\mu }+i\bar{\theta }{\bar{\sigma }}^{\mu }\theta$ it expands as

A category is well-powered when every object has only a set of subobjects. It is well-copowered when every object has only a set of quotient objects.

A continuous subharmonic function on the closure of a bounded domain attains its maximum on the boundary. If it is twice differentiable, the strict condition $\Delta u>0$ rules out an interior maximum immediately because the Hessian matrix there would be negative semidefinite.

Every subgroup $H\le G$ of index two is normal: the two left cosets and the two right cosets are both $H$ and its complement, so they agree.

The index $[G:H]$ is the number of left cosets of $H$ in $G$.

Let an automorphism $\sigma$ of an algebraically closed field $L$ have finite order and fixed field $K$. Artin's fixed-field theorem makes $L/K$ finite cyclic Galois. Every finite extension $M/K$ embeds in $L$: in characteristic zero, choose a primitive generator and send it to a root in $L$. Its image is an intermediate field of a cyclic extension and is therefore Galois and cyclic over $K$.

For $r>1$, choose a prime $\ell \equiv 1(\mathrm{mod}r)$ by Dirichlet's theorem. The cyclic grouphas a subgroup of index $r$, whose fixed field is cyclic Galois of degree $r$ over $\mathbb{Q}$.

A connected graph is unicyclic when it contains exactly one cycle, equivalently when its numbers of vertices and edges agree.

A pseudoforest permits both these components and acyclic tree components.

For a bipartite density operator ${\rho }_{AB}$,

A finite abelian group is indecomposable exactly when it is cyclic of prime-power order. The structure theorem proves necessity. Conversely, every nonzero subgroup of $C_{p^n}$ contains its unique subgroup of order $p$, so two nonzero subgroups cannot be complementary direct summands.

An elementary divisor is a prime power $p^e$ occurring in the cyclic decomposition of a finitely generated torsion module.

Every finitely generated torsion-free module over a principal ideal domain is free. In the structure theorem, torsion-freeness removes every cyclic torsion summand and leaves only the free summand.

The Bessel potential of order $s$ is the Fourier multiplier $(1+|\xi {|}^2{)}^{-s/2}$. Sufficiently high order turns a compactly supported finite-order distribution into a bounded continuous function.