Cauchy's Theorem

Suppose a group has exactly 12 elements. Must it contain an element of order 3? Must it contain an element of order 5? These questions look similar. But their answers differ, and the difference is the entire content of Cauchy's Theorem. 3 divides 12 while 5 does not. The theorem asserts that this arithmetic condition alone forces the existence of the corresponding element. This matters because it is the first genuine bridge. It connects the arithmetic of a group's order — a single natural number — to the group's internal structure. Lagrange's Theorem tells us that the order of every element divides the order of the group. But it gives no converse: knowing that dd divides ∣G∣|G| does not by itself guarantee an element of order dd. Cauchy's Theorem rescues a converse in the one case where it is always true: when dd is prime. This single fact is the seed from which the entire theory of finite groups grows. The big picture is that Cauchy's Theorem is the base case of the The Sylow Theorems. These vastly generalize it by producing not just an element but an entire subgroup of maximal prime-power order. It is also the essential ingredient in the classification of small groups. It plays a key role in the structure theory of p-Groups and Their Properties. It also appears in countless arguments where one needs "an element of order pp" as a foothold. The proof we give is due essentially to James McKay, reworked from Cauchy's original combinatorial argument. It is a beautiful and now-standard application of Group Actions. It also draws on the orbit-counting ideas behind The Class Equation. This article states and proves Cauchy's Theorem in full generality for finite groups. It builds the group-action machinery from scratch. It works two asymmetric examples showing the theorem's mechanics on concrete groups. It also shows precisely why finiteness is indispensable. We do this by exhibiting an infinite group with no element of a given prime order, despite that prime "dividing" nothing sensible. This clarifies exactly where the hypothesis bites.

Key takeaways
  • Cauchy's Theorem: if a finite group GG has order divisible by a prime pp, then GG has an element of order pp.
  • It is a genuine converse to Lagrange's Theorem, but only for prime divisors — the general converse is false.
  • The slickest modern proof (McKay's proof) uses a group action on tuples with product identity, plus counting orbits mod pp.
  • The theorem requires GG finite; without finiteness the statement doesn't even parse the same way, and can fail.
  • It is the prime-order base case that the Sylow theorems generalize to prime-power subgroups.
  • Key idea: count fixed points of a cleverly chosen Z/pZ\mathbb{Z}/p\mathbb{Z}-action modulo pp to force a nontrivial solution to xp=ex^p = e.

Where this lives today

Cauchy's Theorem is the indispensable first rung of finite group classification: it is the base case that The Sylow Theorems generalize, replacing "an element of order pp" with "a whole subgroup of order pkp^k," and those Sylow subgroups drive nearly all structural results about finite groups, including the classification of groups of small order and deep results in the classification of finite simple groups. In p-Groups and Their Properties, Cauchy's Theorem underlies the corollary above characterizing pp-groups purely by element orders, feeding directly into the study of nilpotent groups. Beyond pure group theory, Cauchy's Theorem is used implicitly whenever a finite structure's automorphism group or symmetry group is analyzed for cyclic subgroups: crystallography and chemistry use finite point groups whose prime-order elements (rotations of prime order) are guaranteed exactly by this theorem. In coding theory and cryptography, groups such as (Z/nZ)×(\mathbb{Z}/n\mathbb{Z})^{\times} arising in RSA and Diffie–Hellman rely on knowing which prime orders of elements exist, connecting directly to Cauchy's Theorem via the structure of finite abelian groups. In computational group theory and computer algebra systems, algorithms for finding elements of specified prime order in a permutation or matrix group are direct algorithmic realizations of the existence guaranteed here, often as a subroutine feeding Sylow subgroup construction algorithms.

Ask the tutor: How exactly does Cauchy's Theorem serve as the base case inside the proof of the first Sylow Theorem?

DefinitionElement of Order pp, and the Statement of the Property

Let GG be a group and pp a prime number. An element g∈Gg \in G has order pp if g≠eg \neq e and gp=eg^p = e; equivalently, the cyclic subgroup ⟨g⟩\langle g \rangle has exactly pp elements.

Cauchy's Theorem is the following existence statement: for every finite group GG and every prime pp dividing ∣G∣|G|, there exists an element g∈Gg \in G with gp=eg^p = e and g≠eg \neq e.

Intuition

Lagrange's Theorem is a strong necessary condition: every subgroup's order divides the group's order, so every element's order divides ∣G∣|G|. It says nothing about which divisors actually occur as orders of elements. For composite divisors this silence is justified — A4A_4 has order 1212, and 66 divides 1212, yet A4A_4 has no element of order 66 (its elements have orders 1,2,31, 2, 3 only). So a naive converse to Lagrange is simply false. What makes primes special is a counting trick, not a structural one. We want to solve xp=ex^p = e with x≠ex \neq e. Consider the set of all pp-tuples (x1,…,xp)(x_1, \dots, x_p) of elements of GG whose product is the identity. Fixing the first p−1p-1 entries arbitrarily determines the last entry uniquely (as the inverse of the partial product), so there are exactly ∣G∣p−1|G|^{p-1} such tuples. Cyclically rotating a tuple that multiplies to ee produces another tuple that multiplies to ee — this is the one non-obvious algebraic fact, proved below — so cyclic rotation defines an action of Z/pZ\mathbb{Z}/p\mathbb{Z} on this set of size ∣G∣p−1|G|^{p-1}. Orbits under this action have size 11 or pp (since pp is prime), and a fixed tuple is precisely one where all entries are equal, i.e., a solution to xp=ex^p = e. Since p∣∣G∣p−1p \mid |G|^{p-1}, counting orbits modulo pp forces the number of fixed tuples to be a multiple of pp too. There is always at least one fixed tuple — the all-identity tuple (e,e,…,e)(e, e, \dots, e) — so there must be at least pp of them, meaning at least one nontrivial solution exists.

Figure
Rotating a 5-tuple with product ee: an orbit of size p=5p=5 versus a fixed tuple
Orbit of size p = 5 (rotation acts freely) (a, b, c, d, a⁻¹b⁻¹c⁻¹d⁻¹) (a⁻¹b⁻¹c⁻¹d⁻¹, a, b, c, d) (d, a⁻¹b⁻¹c⁻¹d⁻¹, a, b, c) (c, d, a⁻¹b⁻¹c⁻¹d⁻¹, a, b) (b, c, d, a⁻¹b⁻¹c⁻¹d⁻¹, a) Fixed tuple: orbit of size 1 (g, g, g, g, g) with g⁵ = e g⁵ = e |G|⁴ tuples total, split into orbits of size 5 or 1; so (# fixed tuples) ≡ |G|⁴ ≡ 0 (mod 5), and (e,…,e) is one — so more exist.

Theorem and Proof

TheoremCauchy's Theorem

Let GG be a finite group and let pp be a prime number such that p∣∣G∣p \mid |G|. Then GG contains an element of order pp.

LemmaThe rotation action is well defined

Let pp be a prime and let

X={(x1,x2,…,xp)∈Gp:x1x2⋯xp=e}.X = \{(x_1, x_2, \dots, x_p) \in G^p : x_1 x_2 \cdots x_p = e\}.

Define σ:X→X\sigma : X \to X by cyclically shifting entries:

σ(x1,x2,…,xp)=(x2,x3,…,xp,x1).\sigma(x_1, x_2, \dots, x_p) = (x_2, x_3, \dots, x_p, x_1).

Then σ\sigma is a well-defined bijection X→XX \to X, and it generates an action of the cyclic group Z/pZ\mathbb{Z}/p\mathbb{Z} on XX via k⋅(x1,…,xp)=σk(x1,…,xp)k \cdot (x_1,\dots,x_p) = \sigma^k(x_1,\dots,x_p).

Proof

Step 1: ∣X∣=∣G∣p−1|X| = |G|^{p-1}. A tuple (x1,…,xp)(x_1, \dots, x_p) lies in XX if and only if x1,…,xp−1x_1, \dots, x_{p-1} are arbitrary and

xp=(x1x2⋯xp−1)−1.x_p = (x_1 x_2 \cdots x_{p-1})^{-1}.

This is because the defining equation x1⋯xp−1xp=ex_1 \cdots x_{p-1} x_p = e has, for any fixed choice of x1,…,xp−1x_1,\dots,x_{p-1}, exactly one solution for xpx_p in a group (multiply both sides on the left by (x1⋯xp−1)−1(x_1\cdots x_{p-1})^{-1}). So the map Gp−1→XG^{p-1} \to X sending (x1,…,xp−1)↦(x1,…,xp−1,(x1⋯xp−1)−1)(x_1,\dots,x_{p-1}) \mapsto (x_1,\dots,x_{p-1}, (x_1\cdots x_{p-1})^{-1}) is a bijection, and hence

∣X∣=∣G∣p−1.|X| = |G|^{p-1}.

Step 2: σ\sigma maps XX into XX. Suppose x1x2⋯xp=ex_1 x_2 \cdots x_p = e. We must show x2x3⋯xpx1=ex_2 x_3 \cdots x_p x_1 = e. From x1(x2⋯xp)=ex_1(x_2\cdots x_p) = e we get x2⋯xp=x1−1x_2 \cdots x_p = x_1^{-1}. Multiply on the right by x1x_1:

x2⋯xpx1=x1−1x1=e.x_2 \cdots x_p x_1 = x_1^{-1} x_1 = e.

So σ(x1,…,xp)∈X\sigma(x_1,\dots,x_p) \in X whenever (x1,…,xp)∈X(x_1,\dots,x_p) \in X. This is the key algebraic fact referenced in the intuition: a cyclic rotation of a tuple multiplying to ee again multiplies to ee.

Step 3: σ\sigma is a bijection on XX, and σp=idX\sigma^p = \mathrm{id}_X. The inverse map is the reverse rotation τ(x1,…,xp)=(xp,x1,…,xp−1)\tau(x_1,\dots,x_p) = (x_p, x_1, \dots, x_{p-1}), and the same computation as Step 2 (applied to the tuple (xp,x1,…,xp−1)(x_p, x_1,\dots,x_{p-1}), using that its rotation is (x1,…,xp)∈X(x_1,\dots,x_p) \in X) shows τ\tau maps XX into XX as well. Direct composition gives σ∘τ=τ∘σ=idX\sigma \circ \tau = \tau \circ \sigma = \mathrm{id}_X, so σ\sigma is a bijection with inverse τ\tau. Applying σ\sigma exactly pp times cycles every coordinate back to its starting position, so σp=idX\sigma^p = \mathrm{id}_X.

Step 4: this yields a well-defined group action of Z/pZ\mathbb{Z}/p\mathbb{Z}. Define k‾⋅x=σk(x)\overline{k} \cdot x = \sigma^k(x) for k‾∈Z/pZ\overline{k} \in \mathbb{Z}/p\mathbb{Z} and x∈Xx \in X, choosing any integer representative kk of the class k‾\overline{k}. This is well defined because if k≡k′(modp)k \equiv k' \pmod p then σk=σk−k′σk′=(σp)(k−k′)/pσk′=σk′\sigma^k = \sigma^{k-k'}\sigma^{k'} = (\sigma^p)^{(k-k')/p}\sigma^{k'} = \sigma^{k'} using σp=idX\sigma^p = \mathrm{id}_X from Step 3. The identity of Z/pZ\mathbb{Z}/p\mathbb{Z} acts as σ0=idX\sigma^0 = \mathrm{id}_X, and k1‾⋅(k2‾⋅x)=σk1(σk2(x))=σk1+k2(x)=(k1‾+k2‾)⋅x\overline{k_1}\cdot(\overline{k_2}\cdot x) = \sigma^{k_1}(\sigma^{k_2}(x)) = \sigma^{k_1+k_2}(x) = (\overline{k_1}+\overline{k_2})\cdot x, so the group action axioms hold. ■\blacksquare

∎
Proof

Proof of Cauchy's Theorem, using the Lemma and the Group Actions orbit-counting framework underlying The Class Equation. Let p∣∣G∣p \mid |G| and form XX and the Z/pZ\mathbb{Z}/p\mathbb{Z}-action σ\sigma as in the Lemma. Since Z/pZ\mathbb{Z}/p\mathbb{Z} acting on XX partitions XX into orbits, and the orbit-stabilizer theorem states that each orbit's size divides the order of the acting group, every orbit of this action has size dividing pp. Since pp is prime, every orbit has size exactly 11 or exactly pp. Let F⊆XF \subseteq X be the set of tuples forming orbits of size 11, i.e., fixed points of σ\sigma: tuples with σ(x1,…,xp)=(x1,…,xp)\sigma(x_1,\dots,x_p) = (x_1,\dots,x_p), meaning x1=x2=⋯=xpx_1 = x_2 = \cdots = x_p. A tuple in FF therefore has the form (g,g,…,g)(g,g,\dots,g) with gp=eg^p = e (since the tuple lies in XX, its product gpg^p equals ee). Conversely any such tuple is visibly fixed by σ\sigma and lies in XX. So

F={(g,g,…,g):g∈G, gp=e},F = \{(g,g,\dots,g) : g \in G,\ g^p = e\},

and there is a bijection between FF and the set S={g∈G:gp=e}S = \{g \in G : g^p = e\}. Partition XX into its orbits. Every orbit not contained in FF has size exactly pp. Hence

∣X∣=∣F∣+p⋅(number of orbits of size p).|X| = |F| + p\cdot(\text{number of orbits of size } p).

Rearranging,

∣F∣=∣X∣−p⋅(number of size-p orbits)≡∣X∣(modp).|F| = |X| - p \cdot (\text{number of size-}p\text{ orbits}) \equiv |X| \pmod p.

By Step 1 of the Lemma, ∣X∣=∣G∣p−1|X| = |G|^{p-1}. Since p∣∣G∣p \mid |G|, we have p∣∣G∣p−1p \mid |G|^{p-1} (as p−1≥1p-1 \geq 1 because p≥2p \geq 2; if p−1=0p-1 = 0 we would need p=1p=1, impossible for a prime). Therefore ∣X∣≡0(modp)|X| \equiv 0 \pmod p, and so

∣F∣≡0(modp).|F| \equiv 0 \pmod p.

Now, e∈Se \in S trivially since ep=ee^p = e; equivalently (e,e,…,e)∈F(e,e,\dots,e) \in F. So ∣F∣≥1|F| \geq 1. Combined with p∣∣F∣p \mid |F|, and ∣F∣≥1>0|F| \geq 1 > 0, we conclude ∣F∣≥p≥2|F| \geq p \geq 2. Hence FF contains at least one element besides (e,…,e)(e,\dots,e), i.e., SS contains some g≠eg \neq e with gp=eg^p = e. Such a gg has order dividing pp (since gp=eg^p = e) but order ≠1\neq 1 (since g≠eg \neq e); as pp is prime its only divisors are 11 and pp, so the order of gg is exactly pp. This exhibits the required element of order pp in GG. ■\blacksquare

∎

Proof technique used: group action of a cyclic group of prime order on a combinatorially defined set, combined with the orbit-stabilizer theorem and counting fixed points modulo pp — the McKay proof.

CorollaryA finite group is a pp-group iff its order is a prime power

Let GG be finite and pp prime. If every element of GG has order a power of pp (including 1=p01 = p^0), then ∣G∣|G| is a power of pp.

Proof

We prove the contrapositive. Suppose ∣G∣|G| is not a power of pp. Then some prime q≠pq \neq p divides ∣G∣|G| (by unique factorization of ∣G∣|G| into primes, if no prime other than pp divided ∣G∣|G| then ∣G∣|G| would itself be a power of pp). By Cauchy's Theorem applied to the prime qq, GG has an element of order qq, which is not a power of pp since q≠pq \neq p and qq is prime. This contradicts the hypothesis that every element has order a power of pp. ■\blacksquare

∎

Worked examples

Example 1: G=S4G = S_4, the symmetric group on 4 letters, ∣G∣=24|G| = 24, and p=3p = 3. We know 3∣243 \mid 24, so Cauchy's Theorem guarantees an element of order 33 in S4S_4. Rather than merely trusting the theorem, let's watch McKay's proof produce one, and separately verify directly. Direct verification: take x=(1 2 3)∈S4x = (1\,2\,3) \in S_4, a 3-cycle. Compute x2=(1 3 2)≠ex^2 = (1\,3\,2) \neq e and x3=(1 2 3)3=ex^3 = (1\,2\,3)^3 = e (a 3-cycle cubed returns every element to its start). So xx has order exactly 33, confirming the theorem's conclusion concretely, and there are in fact 88 such 3-cycles in S4S_4 (choose the 3 moved points in (43)=4\binom{4}{3}=4 ways, times 22 cyclic orderings each). Now connect this to the proof mechanism: with p=3p = 3, X={(x1,x2,x3)∈S43:x1x2x3=e}X = \{(x_1,x_2,x_3) \in S_4^3 : x_1x_2x_3 = e\} has size ∣S4∣2=576|S_4|^2 = 576. One fixed tuple is the trivial (e,e,e)(e,e,e). The proof guarantees ∣F∣≡0(mod3)|F| \equiv 0 \pmod 3 and ∣F∣≥1|F|\ge 1, so ∣F∣≥3|F| \ge 3; concretely, FF contains tuples like ((1 2 3),(1 2 3),(1 2 3))((1\,2\,3),(1\,2\,3),(1\,2\,3)) since (1 2 3)3=e(1\,2\,3)^3 = e, and its "partner" fixed tuple from the inverse 3-cycle ((1 3 2),(1 3 2),(1 3 2))((1\,3\,2),(1\,3\,2),(1\,3\,2)), together with (e,e,e)(e,e,e) — already three fixed tuples as predicted, though FF is in fact much larger since every element of order dividing 3 (i.e. ee and all eight 3-cycles) contributes one fixed tuple, giving ∣F∣=9|F| = 9, which is indeed ≡0(mod3)\equiv 0 \pmod 3.

Example 2: G=Z/4Z×Z/3ZG = \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}, ∣G∣=12|G| = 12, and p=2p = 2. This example is deliberately asymmetric — a mixed-order abelian group where we must actually track structure, not just cycle types. Elements are pairs (a,b)(a,b) with a∈Z/4Za \in \mathbb{Z}/4\mathbb{Z}, b∈Z/3Zb \in \mathbb{Z}/3\mathbb{Z}, group operation componentwise. Since 2∣122 \mid 12, Cauchy's Theorem guarantees an element of order 22. Search directly: the order of (a,b)(a,b) is lcm(ord(a),ord(b))\mathrm{lcm}(\mathrm{ord}(a), \mathrm{ord}(b)) since the group is a direct product. We want this lcm to equal 22. Since ord(b)∈{1,3}\mathrm{ord}(b) \in \{1,3\} for b∈Z/3Zb \in \mathbb{Z}/3\mathbb{Z}, and 22 does not divide any lcm involving a factor of 33 unless that factor is 11, we need b=0b = 0 (order 1) and ord(a)=2\mathrm{ord}(a) = 2. In Z/4Z\mathbb{Z}/4\mathbb{Z}, the element a=2a = 2 has order 22 since 2+2=4≡02 + 2 = 4 \equiv 0 but 2≢02 \not\equiv 0. So g=(2,0)g = (2, 0) has order 22: indeed g+g=(4,0)=(0,0)=eg + g = (4,0) = (0,0) = e and g≠eg \neq e. This is the unique element of order 22 in GG, matching the fact that Z/4Z\mathbb{Z}/4\mathbb{Z} has a unique element of order 22 and Z/3Z\mathbb{Z}/3\mathbb{Z} contributes nothing to order-2 elements. Note also what Cauchy's Theorem does not promise here: 6∣126 \mid 12 as well, yet one can check every element of GG has order in {1,2,3,4}\{1,2,3,4\} — for instance (1,1)(1,1) has order lcm(4,3)=12\mathrm{lcm}(4,3)=12, and no element has order exactly 66 (that would need lcm(ord(a),ord(b))=6\mathrm{lcm}(\mathrm{ord}(a),\mathrm{ord}(b)) = 6, impossible since ord(a)∈{1,2,4}\mathrm{ord}(a) \in \{1,2,4\} never contributes a factor of 33 needed alongside a proper compatibility, and checking all 1212 elements confirms orders are exactly 1,2,3,4,4,4,4,12,12,12,121,2,3,4,4,4,4,12,12,12,12-type patterns, none equal to 66). This reinforces that the prime hypothesis on pp in Cauchy's Theorem is essential — see the counterexample below for the general lesson.

CounterexampleWhy "finite" cannot be dropped

Cauchy's Theorem requires GG to be a finite group. Drop finiteness, and the conclusion can fail entirely, even in a form that makes sense to ask. Consider G=ZG = \mathbb{Z}, the additive group of integers, which is infinite. Let p=2p = 2. Is there an element g∈Zg \in \mathbb{Z}, g≠0g \neq 0, with 2g=02g = 0 (writing the group additively, so "order 22" means 2g=02g = 0 and g≠0g \neq 0)? In Z\mathbb{Z}, 2g=02g = 0 forces g=0g = 0 since Z\mathbb{Z} has no zero divisors under multiplication by integers and division by 22 is injective on Z\mathbb{Z}. So Z\mathbb{Z} has no element of order 22 — nor of any finite order other than the identity — despite 22 being "compatible" with the group in every naive sense one might imagine transferring from the finite case. Where does the proof break down? The entire argument rested on ∣X∣=∣G∣p−1|X| = |G|^{p-1} being a specific finite number congruent to 00 mod pp; for infinite GG, XX is infinite and "counting orbits mod pp" is meaningless — there is no notion of ∣X∣  p|X| \bmod p for an infinite set. The orbit-counting mechanism that forces ∣F∣≥p|F| \geq p simply has no infinite analogue without extra structure (e.g., a well-behaved measure or additional finiteness assumptions). This is not a minor technicality: it is the entire engine of the proof, and it stalls completely once GG is infinite. A subtler moral: even some infinite groups do contain elements of every prime order (e.g. Q/Z\mathbb{Q}/\mathbb{Z} has elements of every order), so infiniteness alone doesn't force failure — it only removes the guarantee. The theorem's hypothesis of finiteness is precisely what supplies the counting argument with a finite, well-defined quantity to reduce modulo pp.

Scope, edge cases & common traps

SituationWhat Cauchy's Theorem saysCommon error
p∣∣G∣p \mid \lvert G \rvert, pp prime, GG finiteGuarantees an element of order exactly ppForgetting pp must be prime; theorem is false for composite dd (e.g. A4A_4, d=6d=6)
p∤∣G∣p \nmid \lvert G \rvertSays nothing; typically no element of order pp (consistent with Lagrange)Assuming the theorem's silence means such an element cannot exist — it merely isn't guaranteed by this theorem (Lagrange actually forbids it)
GG infiniteTheorem does not applyMisapplying the "mod pp" counting argument to infinite XX
GG trivial, ∣G∣=1\lvert G\rvert = 1No prime divides 11, vacuously no hypothesis to satisfy—
Multiple primes dividing ∣G∣|G|Theorem applies separately to each prime factor, giving one element of order pp for eachConflating existence of elements of order pp and order qq with existence of an element of order pqpq

The mathematician behind it

Origins · 1 minAugustin-Louis Cauchy and how this was discovered
AC
Augustin-Louis Cauchy
1789–1857 · Early 19th-century French mathematical analysis and the birth of group theory

Cauchy was one of the most prolific mathematicians in history, foundational to rigorous analysis (limits, continuity, convergence) and an early pioneer of permutation group theory, decades before the abstract group axioms were formalized by Cayley and others. His 1845 paper proving what we now call Cauchy's Theorem worked directly with permutation groups, using a combinatorial argument on cycle structures rather than the abstract group-action language used today. The modern short proof presented here, via a Z/pZ\mathbb{Z}/p\mathbb{Z}-action on tuples, is due to James H. McKay (1959), who recast Cauchy's result in a strikingly economical way that is now the standard textbook proof.

History
- 1789 — Cauchy born in Paris, into a turbulent revolutionary France.
- 1815–1830 — Cauchy establishes rigorous foundations of real and complex analysis while also studying permutations and determinants.
- 1845 — Cauchy publishes the theorem bearing his name, for permutation groups, using cycle-structure combinatorics.
- 1854 — Cayley's abstract definition of a group appears, later allowing Cauchy's result to be restated for all finite groups.
- 1959 — James McKay publishes the now-standard slick proof via a group action on tuples of product identity.
Does Cauchy's Theorem give a unique element of order pp?

No, uniqueness is never claimed or true in general. The theorem only guarantees existence of at least one element of order pp. In S4S_4 with p=3p=3 there are eight elements of order 33 (all the 3-cycles), not one. The proof itself shows ∣F∣≥p|F| \geq p, meaning at least p−1p-1 nontrivial solutions to gp=eg^p=e typically exist (grouped in orbits under conjugation-like symmetry in the abstract counting), and often far more depending on the group's structure. Counting the exact number of elements of order pp requires more refined tools such as Sylow theory or explicit structure theorems, not Cauchy's Theorem alone.

Is Cauchy's Theorem true for infinite groups if we only ask about elements of finite order?

Not in general. The theorem as stated requires GG finite so that ∣G∣|G| is a number a prime can divide. If GG is infinite one can still ask "does GG have an element of order pp?" but there is no longer any implication from an arithmetic hypothesis like "pp divides something," since ∣G∣|G| isn't a finite number to begin with. Some infinite groups (like Q/Z\mathbb{Q}/\mathbb{Z}) have elements of every prime order; others (like Z\mathbb{Z}) have none of any finite order at all. The theorem simply has no infinite analogue without extra hypotheses replacing "p∣∣G∣p \mid |G|."

How does Cauchy's Theorem relate to the Sylow theorems?

Cauchy's Theorem is exactly the special case k=1k=1 of the first Sylow theorem, which asserts that if pkp^k divides ∣G∣|G| then GG has a subgroup of order pkp^k. When k=1k=1, a subgroup of order pp is cyclic (as any group of prime order is cyclic, generated by any non-identity element), so having "a subgroup of order pp" is equivalent to having "an element of order pp." The full The Sylow Theorems use substantially more machinery (often assuming Cauchy's Theorem as a lemma, or reproving it along the way) to handle all prime-power divisors, not just the prime itself.

Can the McKay proof be adapted to show there are exactly p−1p-1 nontrivial elements of order pp, or some other precise count?

The proof only shows ∣F∣≡0(modp)|F| \equiv 0 \pmod p and ∣F∣≥1|F| \geq 1, hence ∣F∣≥p|F| \geq p. It does not pin down ∣F∣|F| exactly, because the number of size-pp orbits in X∖FX \setminus F is not controlled by the argument. In fact different groups of the same order and same prime pp can have very different counts of elements of order pp: cyclic groups of order pp have exactly p−1p - 1 such elements, while larger groups can have many more. Precise counts require group-specific analysis, character theory, or direct combinatorics beyond what Cauchy's Theorem provides.

Why is the tuple set XX constructed with product equal to ee, rather than some other condition?

The condition "product equals ee" is precisely what makes rotation well-defined as a self-map of XX (Step 2 of the Lemma), because it lets us transfer one factor from the front to the back using the group axioms without changing the product. It is also exactly the condition that turns a fixed tuple (g,g,…,g)(g,g,\dots,g) into the equation gp=eg^p = e, which is what we want to solve. Any other defining condition for XX would generally fail to be rotation-invariant, breaking the group action structure the whole proof depends on.

Does Cauchy's Theorem hold for infinite direct products of finite groups, like an infinite product of copies of Z/2Z\mathbb{Z}/2\mathbb{Z}?

The group itself (an infinite direct product) is infinite, so Cauchy's Theorem's hypotheses are not met and it does not directly apply. However, such a group may still happen to have elements of a given prime order — for instance, an infinite direct product of copies of Z/2Z\mathbb{Z}/2\mathbb{Z} has every non-identity element of order exactly 22. This is a coincidence of that specific construction, not a consequence of the theorem, and other infinite groups (like Z\mathbb{Z}) have no elements of finite order at all, illustrating that finiteness cannot be dropped without losing the guarantee entirely.

Before this

Where this leads