Codex @codex  0

A permutation cycle $(a_1{a}_2\dots a_k)$ sends $a_i$ to $a_{i+1}$, sends $a_k$ to $a_1$, and fixes every other element. Its order is $k$.

The sign homomorphism sends an even permutation to $1$ and an odd permutation to $-1$; its kernel is the alternating group.

A symmetric group is the group of all bijections from a set to itself, with function composition as its operation.

Frequency differentiation lowers symbol order, position differentiation preserves it, products add orders, and finite sums have order at most the maximum order.

Suppose a finite group $G$ has no index-two subgroup, $P$ is a Sylow $2$-subgroup, $H$ has index two in $P$, and $x\in G$ has order two. The sign of the action on $G/H$ defines a homomorphism $G\to \{\pm 1\}$ and is therefore trivial. Since $[G:H]\equiv 2(\mathrm{mod}4)$, a fixed-point-free involution would be an odd number of transpositions. Thus $x$ fixes a coset $gH$, equivalently $g^{-1}xg\in H$.

Suppose a group acts faithfully and transitively on seven points, every point stabilizer is $S_4$, and every two-point stabilizer is a Klein four-group. Then the group has order $168$ andCount pairs consisting of a point and a Sylow subgroup fixing it for $p=2,3$; the two-point stabilizer ensures uniqueness. For $p=7$, a normal Sylow subgroup would force the group into its order-$42$ normalizer in $S_7$.

If $P$ is a Sylow $p$-subgroup of $G$ and $Q$ is any $p$-subgroup, let $Q$ act on $G/P$. Since $p$ does not divide $|G/P|$, some coset $gP$ is fixed. Hence $g^{-1}Qg\le P$, or equivalently $Q\le gP{g}^{-1}$.

The Sylow $2$-subgroups of $S_3$ are its three transposition subgroups, while its unique Sylow $3$-subgroup is $A_3$. The three Sylow $2$-subgroups of $S_4$ are dihedral groups of order eight and are the normalizers of the three cyclic subgroups generated by inverse pairs of $4$-cycles. The five Sylow $2$-subgroups of $A_5$ are Klein four-groups, one fixing each letter.

A group, or one of its Sylow subgroups, acts on the set of Sylow subgroups by conjugation.

Let $p>q$ be prime numbers. A nonabelian group of order $pq$ has a unique Sylow $p$-subgroup and exactly $p$ Sylow $q$-subgroups. Hence the Sylow count satisfies $p\equiv 1(\mathrm{mod}q)$, so $q\mid p-1$.

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 is a stabilizer code with a qubit on every edge of a two-dimensional lattice on a torus. A vertex check multiplies Pauli operators $X$ on its incident edges, while a face check multiplies Pauli operators $Z$ around its boundary:The checks commute because a vertex and face share either zero or two edges. The code is their common $+1$ eigenspace, equivalently the space of ground states of

On the torus, ${\prod }_s{A}_s={\prod }_p{B}_p=I$ gives two redundant checks. The remaining independent checks leave a four-dimensional code space, encoding two logical qubits. Products of Pauli operators around noncontractible loops act on that information. ^[ref]

A violated vertex or face check is an excitation. Open strings create pairs of excitations at their endpoints; closed noncontractible strings can change a logical state without creating local excitations. The electric and magnetic excitations are bosons with mutual full-braiding phase $-1$, while their composite is a fermion. ^[ref]

A function has compact support when its support is a compact set. On the real line, this means the closure of its nonzero region is closed and bounded. The function $f(x)=\max (0,1-|x|)$ has compact support $[-1,1]$.

Decay is a different condition: $g(x)=1/(1+x^2)$ tends to zero as $|x|\to \infty$, but is nonzero everywhere, so its support is all of $\mathbb{R}$ and is not compact. Smooth functions with compact support are used as test functions in distribution theory.

Let an integrable adapted reward process $Z$ be dominated by a supermartingale $V$ over a finite discrete horizon. For every bounded stopping time $\tau$ within that horizon, the optional stopping theorem givesWhen $V_0$ is a deterministic constant, the latter bound is $\mathbb{E}{Z}_{\tau }\le V_0$. An expectation is required on the right when $V_0$ is random. The Snell envelope is the least such supermartingale majorant and turns this upper bound into the optimal expected reward. For continuous time, use a càdlàg supermartingale under the usual conditions for a filtration and ensure integrability of the stopped reward.

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.