Subgroups of Finite Index and Division Subrings

Q: Why is the projective space defined with a right action?

So that the scalar action and the coordinates sit on the same side. In the injectivity computation one arrives at k = k'k'' with k'' multiplying from the right; if V were a left space acted on by right multiplication, the two sides would interfere and the coefficient comparison would fail. Over a commutative field the distinction evaporates, which is why it is rarely mentioned in classical projective geometry.

Q: Does (13.24) mean D^* has no subgroups of finite index at all?

No. It says nothing about subgroups that are not of the form K^* for a division subring K. The theorem is specifically about division subrings, and the proof relies on K being closed under addition — which an arbitrary subgroup of D^* is not.

Q: How does this compare with the situation for fields?

It is the same statement, and for fields it is classical: an infinite field L with a proper subfield K has [L^*:K^*] infinite, because P^1(K) already has |K|+1 points. The content of (13.22) is that no commutativity is used anywhere, so the conclusion transfers to division rings without modification.

Q: Why does (13.26) need Wedderburn's Little Theorem?

Because (13.25) requires D to be infinite before it can conclude that the class is infinite. Noncentrality of a makes D noncommutative, and it is Wedderburn's theorem that upgrades noncommutative to infinite. Without it, a hypothetical finite noncommutative division ring would have finite conjugacy classes and the argument would stall.

Q: What is the exact size of a conjugacy class, not just its finiteness?

It is [D^* : C_D(a)^*], by orbit–stabiliser. Scott's theorem sharpens (13.26) to say that for noncentral a this index has the same cardinality as D itself, so the class is as large as the ring allows.

Q: Can (13.23) really be applied to sets that are not subspaces?

Yes — the proof only needs 1 and one element v_2 K inside V, plus v_2 k ∈ V and 1 + v_2 k ∈ V. Any subset closed under right multiplication by K and under addition of elements of K qualifies. This flexibility is what lets the corollary be applied to orbit sets that arise inside a larger ring without a natural module structure.

Executive Summary

How far apart are a division ring and a proper division subring? The answer is: always infinitely far, unless everything in sight is finite. For division rings $K ⊊ D$ , the group index $[D^{*} : K^{*}]$ is finite if and only if $D$ is finite — in which case, by Wedderburn, $D$ is a finite field.

The mechanism is geometric and completely elementary. Give $D$ the structure of a right $K$ -vector space; the coset space $D^{*} / K^{*}$ is then the projective space $P (D)$ . A projective space of dimension at least one contains an affine line, and that line is a faithful copy of the scalar division ring. So $P (D)$ is at least as big as $K$ .

$P (V)$ The object

$\dim \geq 2$ Hypothesis in (13.22)

$K ↪ P (V)$ The injected line

$\infty$

[D^{*} : K^{*}]

when

D

is infinite

Overview

Let $V$ be a right vector space over a division ring $K$ . The group $K^{*}$ acts on $V^{*} = V ∖ {0}$ by right multiplication, freely: if $v k = v$ with $v \neq 0$ then $k = 1$ . The orbit space is written

P (V) = V^{*} / K^{*} = {v K^{*} : v \in V^{*}},

(13.22a)

the projective space of $V$ : its points are the one-dimensional subspaces of $V$ .

Freeness of the action is what makes the counting exact. When $K$ is finite, every orbit has exactly $| K^{*} |$ elements, so $| V^{*} | = | P (V) | \cdot | K^{*} |$ — no orbit-stabiliser bookkeeping is required.

In commutative projective geometry this is the observation that a projective space of positive dimension contains a line, and a line over $K$ has $| K | + 1$ points. Nothing in the argument uses commutativity, so it transfers verbatim to division rings — and the transfer is the entire content of $(13.22)$ .

Applied with $V = D$ and $K$ a proper division subring, it yields $(13.24)$ ; applied to conjugation orbits it yields $(13.25)$ and Herstein's theorem that a noncentral element has infinitely many conjugates.

Learning Objectives

Define $P (V) = V^{*} / K^{*}$ and verify that the $K^{*}$ -action is free.
Construct the injection $K \to P (V)$ and prove injectivity by comparing coordinates.
State $(13.22)$ with the hypothesis $\dim (V_{K}) \geq 2$ and explain why dimension one is excluded.
Weaken the hypothesis to a subset $V ⊋ K$ with $V K \subseteq V$ and $V + K \subseteq V$ .
Deduce $(13.24)$ : $[D^{*} : K^{*}] < \infty$ if and only if $D$ is finite.
Identify the $D$ -conjugates of $a$ with $D^{*} / K^{*}$ for $K = D \cap C_{E} (a)$ , and deduce $(13.26)$ .

Definitions

Definition(13.22a)Projective space over a division ring

For a right $K$ -vector space $V$ over a division ring $K$ , let $K^{*}$ act on $V^{*} = V ∖ {0}$ by $v \cdot k = v k$ . The orbit space $P (V) : = V^{*} / K^{*}$ is the **projective space associated with $V$ **. Its points correspond bijectively to the one-dimensional right $K$ -subspaces of $V$ , and $\dim P (V) = \dim (V_{K}) - 1$ .

$V^{*}$: $V ∖ {0}$ . For $V = D$ a division ring this is the multiplicative group $D^{*}$ .
Free action: $v k = v$ with $v \neq 0$ implies $k = 1$ , so every orbit is a faithful copy of $K^{*}$ .
$[D^{*} : K^{*}]$: The number of right cosets $d K^{*}$ , which is exactly $| P (D) |$ when $D$ is viewed as a right $K$ -vector space.
$K$ -independent vectors: $v_{1}, v_{2}$ with $v_{1} k_{1} + v_{2} k_{2} = 0$ forcing $k_{1} = k_{2} = 0$ . Available as soon as $\dim (V_{K}) \geq 2$ .
$C_{E} (a)$: The centralizer of $a$ in $E$ , a division subring. In $(13.25)$ the relevant object is $K = D \cap C_{E} (a)$ .

Vector spaces here are right vector spaces so that scalars multiply on the same side as the group acts; the left-handed statement is identical after passing to the opposite ring.

Core Concepts

The line, and why it is injective

Choose $K$ -independent $v_{1}, v_{2} \in V$ and consider the family of points $(v_{1} + v_{2} k) K^{*}$ for $k \in K$ . Geometrically these are the points of a projective line other than the point $v_{2} K^{*}$ — an affine line. Suppose two parameters give the same point:

v_{1} + v_{2} k = (v_{1} + v_{2} k') k'' = v_{1} k'' + v_{2} (k' k'') for some k'' \in K^{*} .

(13.22b)

Independence of $v_{1}, v_{2}$ lets the coefficients be compared: $1 = k''$ from the $v_{1}$ -coordinate, and then $k = k' k'' = k'$ . So the parametrisation is injective and $| P (V) | \geq | K |$ .

Why dimension at least two

What is $P (V)$ when $\dim (V_{K}) = 1$ ?

A single point

V = v K

and

V^{*} = v K^{*}

is one orbit. So

P (V)

is finite regardless of how large

K

is — the theorem is false without

\dim \geq 2

Dimension 0

V = 0

V^{*}

empty,

P (V)

empty. Also excluded.

Dimension at least 2Two independent vectors exist, the line embeds, and

| P (V) | \geq | K |

. This is the hypothesis of

(13.22)

From vector spaces to arbitrary closed subsets

The proof uses very little about $V$ : only that $v_{1} = 1$ and some $v_{2} \notin K$ lie in $V$ , and that $v_{2} k$ and $1 + v_{2} k$ stay in $V$ . So the hypothesis " $V$ is a $K$ -subspace" can be replaced by

K ⊊ V \subseteq R, V \cdot K \subseteq V, V + K \subseteq V,

(13.23a)

Enough for the injection $k \mapsto (1 + v_{2} k) K^{*}$ to be defined, with $v_{2}$ any element of $V ∖ K$ .

Independence of $1$ and $v_{2}$ over $K$ is automatic: if $1 \cdot k_{1} + v_{2} k_{2} = 0$ with $k_{2} \neq 0$ , then $v_{2} = - k_{1} k_{2}^{- 1} \in K$ , contrary to choice.

Key Results

Theorem(13.22)Finiteness of a projective space

Let $V$ be a right vector space over a division ring $K$ with $\dim (V_{K}) \geq 2$ . Then $P (V) = V^{*} / K^{*}$ is finite if and only if $V$ is finite — in which case $K$ is finite too.

Proof

**( $\Leftarrow$ )** If $V$ is finite then $V^{*}$ is finite and so is any quotient of it.

**( $\Rightarrow$ )** Since $\dim (V_{K}) \geq 2$ choose $K$ -independent $v_{1}, v_{2} \in V$ and define

A : K ⟶ P (V), A (k) = (v_{1} + v_{2} k) K^{*} .

(13.22c)

$v_{1} + v_{2} k \neq 0$ by independence, so $A$ is well defined.

If $A (k) = A (k')$ then $(13.22 b)$ holds for some $k'' \in K^{*}$ ; comparing $v_{1}$ -coordinates gives $k'' = 1$ , and then the $v_{2}$ -coordinates give $k = k'$ . So $A$ is injective and $| K | \leq | P (V) |$ .

Assume $P (V)$ is finite. Then $K$ is finite by the above. The action of $K^{*}$ on $V^{*}$ is free, so every orbit has exactly $| K^{*} |$ elements and $| V^{*} | = | P (V) | \cdot | K^{*} |$ is finite. Hence $V$ is finite.

Corollary(13.23)Brauer–Faith form

Let $K$ be a division subring of a ring $R$ and let $V$ satisfy $K ⊊ V \subseteq R$ with $V \cdot K \subseteq V$ and $V + K \subseteq V$ — in particular this holds when $V$ is a right $K$ -subspace of $R_{K}$ properly containing $K$ . Let $K^{*}$ act on $V^{*} = V ∖ {0}$ by right multiplication. Then $V^{*} / K^{*}$ is finite if and only if $V$ is finite.

Proof

Pick $v_{2} \in V ∖ K$ , which exists because $K ⊊ V$ . As noted above, $1$ and $v_{2}$ are right $K$ -independent in $R$ . The closure hypotheses put $v_{2} k \in V$ and hence $1 + v_{2} k \in V$ for all $k \in K$ , so the map $k \mapsto (1 + v_{2} k) K^{*}$ takes values in $V^{*} / K^{*}$ and is injective by the coordinate comparison of $(13.22 b)$ . Finiteness of $V^{*} / K^{*}$ therefore forces $K$ finite, and freeness of the action gives $| V^{*} | = | V^{*} / K^{*} | \cdot | K^{*} | < \infty$ . The converse is immediate.

Corollary(13.24)Index of a proper division subring

Let $K ⊊ D$ be division rings. Then $[D^{*} : K^{*}] < \infty$ if and only if $D$ is finite.

Proof

Apply $(13.23)$ with $R = V = D$ ; the orbits of $K^{*}$ acting on $D^{*}$ by right multiplication are precisely the right cosets $d K^{*}$ , so $| D^{*} / K^{*} | = [D^{*} : K^{*}]$ . The corollary gives finiteness of this index if and only if $D$ is finite.

Combining with Wedderburn's Little Theorem: the only way a proper division subring can have finite index is for $D$ to be a finite field. For every infinite division ring and every proper division subring, the index is infinite.

Corollary(13.25)Counting conjugates

Let $D$ be a division subring of a division ring $E$ and let $a \in E$ . The group $D^{*}$ acts on the set of $D$ -conjugates ${d a d^{- 1} : d \in D^{*}}$ , with isotropy subgroup $K^{*}$ at $a$ , where $K = D \cap C_{E} (a)$ ; so the set of $D$ -conjugates is in bijection with $D^{*} / K^{*}$ . If $D$ is infinite then either $a$ has exactly one $D$ -conjugate — equivalently $D \subseteq C_{E} (a)$ — or it has infinitely many.

Proof

The stabiliser of $a$ consists of those $d \in D^{*}$ with $d a d^{- 1} = a$ , i.e. $d \in D \cap C_{E} (a) = K$ ; note $K$ is a division subring of $D$ , being an intersection of two division subrings of $E$ . The orbit–stabiliser correspondence gives the stated bijection.

If $K = D$ then $D \subseteq C_{E} (a)$ and the orbit is ${a}$ . Otherwise $K ⊊ D$ and, $D$ being infinite, $(13.24)$ gives $[D^{*} : K^{*}] = \infty$ , so the orbit is infinite.

Theorem(13.26)Herstein: noncentral elements have infinitely many conjugates

Let $D$ be a division ring and $a \in D ∖ Z (D)$ . Then $a$ has infinitely many conjugates $x a x^{- 1}$ , $x \in D^{*}$ .

Proof

Since $a$ is noncentral, $D$ is noncommutative, so $D$ is infinite by Wedderburn's Little Theorem $(13.1)$ . Apply $(13.25)$ with $E = D$ : here $K = D \cap C_{D} (a) = C_{D} (a)$ , and $C_{D} (a) \neq D$ precisely because $a \notin Z (D)$ . Hence $K ⊊ D$ and the conjugacy class is infinite.

Remark—Scott's sharpening

More is true: in any division ring $D$ , the conjugacy class of a noncentral element has cardinality equal to $| D |$ . This refinement is due to W. Scott; $(13.26)$ records only that the class is infinite.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Three ideas do all the work, and each is transferable.

Turn an index into a geometry. A coset space $D^{*} / K^{*}$ is an orbit space, and orbit spaces of scalar actions on vector spaces are projective spaces. Once the index is recognised as $P (D)$ , geometric intuition applies.
Find a line. Every projective space of positive dimension contains a copy of the affine line ${v_{1} + v_{2} k}$ , which is parametrised faithfully by the scalars. Lower bounds on the size of a projective space always come from lines.
Use freeness for exact counting. A free action turns $| V^{*} | = | P (V) | \cdot | K^{*} |$ into an identity rather than an inequality, so finiteness propagates in both directions without loss.
Read a conjugacy class as a coset space. Orbit–stabiliser identifies the class of $a$ with $D^{*} / K^{*}$ for $K$ the centralizer, converting a counting question about conjugates into an index question.

Regard $D$ as a right $K$ -space

K ⊊ D

gives

\dim (D_{K}) \geq 2

Identify $D^{*} / K^{*}$ with $P (D)$ The right cosets are exactly the

K^{*}

-orbits.

Inject the scalars

k \mapsto (1 + v_{2} k) K^{*}

for any

v_{2} \in D ∖ K

Conclude $K$ is finiteIf the index is finite,

| K | \leq [D^{*} : K^{*}] < \infty

Count exactlyFreeness gives

| D^{*} | = [D^{*} : K^{*}] \cdot | K^{*} |

, so

D

is finite.

Apply WedderburnA finite division ring is a field, so the degenerate case is entirely commutative.

Worked Example

$ℂ$ inside $ℍ$ : the index is the Riemann sphere

Take $D = ℍ$ and $K = ℂ = ℝ + ℝ i$ , a proper division subring. As a right $ℂ$ -vector space, $ℍ = ℂ \oplus j ℂ$ , of dimension $2$ — the hypothesis of $(13.22)$ holds with $v_{1} = 1$ , $v_{2} = j$ .

P (ℍ) = ℍ^{*} / ℂ^{*} ≅ ℙ^{1} (ℂ),

(E.1)

the complex projective line — the Riemann sphere, of cardinality $2^{ℵ_{0}}$ .

The injection of $(13.22 c)$ is explicit: $k \mapsto (1 + j k) ℂ^{*}$ . If $(1 + j k) ℂ^{*} = (1 + j k') ℂ^{*}$ then $1 + j k = c + j (k' c)$ for some $c \in ℂ^{*}$ ; comparing components in $ℂ \oplus j ℂ$ gives $c = 1$ and $k = k'$ . So $[ℍ^{*} : ℂ^{*}]$ is at least $| ℂ |$ — certainly infinite, exactly as $(13.24)$ requires for the infinite division ring $ℍ$ .

The same count as a conjugacy class

Now run $(13.25)$ with $E = D = ℍ$ and $a = i$ . Its centralizer is $C_{ℍ} (i) = ℂ$ , so the conjugacy class of $i$ is in bijection with $ℍ^{*} / ℂ^{*}$ . Independently, conjugation by a unit quaternion acts on the pure quaternions as a rotation of $ℝ^{3}$ , so

{x i x^{- 1} : x \in ℍ^{*}} = {pure quaternions of norm 1} = S^{2} .

(E.2)

The two descriptions agree: $ℙ^{1} (ℂ)$ and $S^{2}$ are the same set. The abstract bijection of $(13.25)$ is, in this instance, the classical identification of the Riemann sphere with the unit sphere — and it certifies $(13.26)$ , since $i$ is noncentral and has continuum many conjugates.

The degenerate finite case

Take $K = 𝔽_{2} ⊊ D = 𝔽_{4}$ . Here $\dim (D_{K}) = 2$ , and

[D^{*} : K^{*}] = \frac{| 𝔽_{4}^{*} |}{| 𝔽_{2}^{*} |} = \frac{3}{1} = 3 = | ℙ^{1} (𝔽_{2}) |,

(E.3)

a finite index — and indeed $D$ is finite. Consistent with $(13.24)$ , and with Wedderburn: $D$ is a field. Note $| P (D) | = 3 \geq | K | = 2$ , the inequality of the method section, holding with room to spare.

Comparison and Classification

Index of $K^{*}$ in $D^{*}$ across the standard cases
$K \subseteq D$	$\dim (D_{K})$	$[D^{} : K^{}]$	Consistent with (13.24)?
$K = D$	$1$	$1$	not applicable — hypothesis is $K ⊊ D$
$𝔽_{2} ⊊ 𝔽_{4}$	$2$	$3$	yes, $D$ finite
$𝔽_{q} ⊊ 𝔽_{q^{n}}$	$n$	$(q^{n} - 1) / (q - 1)$	yes, $D$ finite
$ℝ ⊊ ℂ$	$2$	infinite	yes, $D$ infinite
$ℂ ⊊ ℍ$	$2$	infinite	yes, $D$ infinite
$Z (D) ⊊ D$ , $D$ infinite	$\geq 4$	infinite	yes

The third row is worth pausing on: for finite fields the index is exactly the number of points of $ℙ^{n - 1} (𝔽_{q})$ , which is the same expression $(q^{n} - 1) / (q - 1)$ that appears as a class-size in the proof of Wedderburn's Little Theorem. The two occurrences have the same origin — both count orbits of a multiplicative group acting freely.

Which hypotheses each statement needs
	$\dim \geq 2$ or $K ⊊ V$	Free action	Wedderburn $(13.1)$	Element noncentral
(13.22)	yes	yes	no	no
(13.23)	yes	yes	no	no
(13.24)	yes	yes	only for the final remark	no
(13.25)	yes, when $K \neq D$	yes	no	no
(13.26)	yes	yes	yes	yes

Which hypotheses each statement needs

Relationship Map

$(13.22)$ — projective spaces are big

| K | \leq | P (V) |

whenever

\dim (V_{K}) \geq 2

$(13.23)$ — the ring-theoretic formHypotheses weakened to

V K \subseteq V

V + K \subseteq V

K ⊊ V

$(13.24)$ — indices of division subrings

[D^{*} : K^{*}] < \infty iff D

finite

$(13.25)$ — counting conjugatesOne conjugate, or infinitely many

$(13.26)$ — HersteinNoncentral

\Rightarrow

infinitely many conjugates

The chain is strictly linear, and only the last link imports anything from outside: $(13.26)$ needs Wedderburn's Little Theorem to know that a noncommutative division ring is infinite. Everything above it is pure counting.

Applications and Industry Use

Applications here means where this structure is used — inside mathematics and in the engineering and computing disciplines that consume it.

Finite geometry

Counting points of projective spaces

The exact count $| V^{*} | = | P (V) | \cdot | K^{*} |$ is the standard derivation of $| ℙ^{n} (𝔽_{q}) | = (q^{n + 1} - 1) / (q - 1)$ , the basic parameter of projective codes, arcs and caps in coding theory and combinatorial design.

Coding theory

Projective codes

Simplex and Hamming codes are defined by taking one representative from each point of $ℙ^{n} (𝔽_{q})$ . The freeness of the scalar action is exactly what makes the column set well defined up to scaling.

Group theory

Subgroups of finite index

$(13.24)$ says $D^{*}$ has no proper subgroup of finite index arising from a division subring when $D$ is infinite — a strong constraint used when analysing normal and finite-index subgroups of linear groups over division rings.

Internal to algebra

Honest summary

For division ring theory itself the value is structural: it forbids the finite-index arguments that are routine in finite group theory, and forces conjugacy classes to be large.

Standards and Notation

Standards here covers notation, symbol and markup standards, and reference implementations, rather than material or design codes.

Projective space

P (V)

here;

ℙ (V)

and

ℙ^{n} (K)

are equally standard

Dimension convention

\dim ℙ^{n} (K) = n = \dim (V_{K}) - 1

Side conventionRight vector spaces here, so scalars act on the right; left conventions require

K^{op}

Index notation

[D^{*} : K^{*}]

for the number of right cosets

Point counts

| ℙ^{n} (𝔽_{q}) | = (q^{n + 1} - 1) / (q - 1)

SoftwareGAP and Magma provide projective spaces over finite fields directly; noncommutative coordinate rings are not supported by default

Failure Modes and Common Mistakes

Do not apply $(13.23)$ to a subset $V$ that fails $V + K \subseteq V$ ; the element $1 + v_{2} k$ must lie in $V$ for the injection to exist.
Do not assume $P (V)$ finite implies $\dim (V_{K})$ finite without also knowing $K$ is finite — the argument produces both, in that order.
Do not conflate $(13.26)$ with the statement that all conjugacy classes are infinite: central elements have exactly one conjugate, themselves.

Quick Reference

Projective space

P (V) = V^{*} / K^{*}

, right action

Freeness

v k = v

v \neq 0

\Rightarrow k = 1

; orbits have size

| K^{*} |

The line

k \mapsto (v_{1} + v_{2} k) K^{*}

is injective for independent

v_{1}, v_{2}

(13.22)

\dim (V_{K}) \geq 2

P (V)

finite

iff V

finite

(13.23)Same with

K ⊊ V

V K \subseteq V

V + K \subseteq V

(13.24)

K ⊊ D

[D^{*} : K^{*}] < \infty iff D

finite

(13.25)

D

infinite: one

D

-conjugate of

a

, or infinitely many

(13.26)

a \notin Z (D) \Rightarrow

infinitely many conjugates

Counting identities used on this page
Identity	Where it comes from
$\| K \| \leq \| P (V) \|$	The injected affine line, $(13.22 c)$
$\| V^{} \| = \| P (V) \| \cdot \| K^{} \|$	Freeness of the scalar action
$\| {d a d^{- 1}} \| = [D^{} : K^{}]$	Orbit–stabiliser, $K = D \cap C_{E} (a)$
$\| ℙ^{n - 1} (𝔽_{q}) \| = (q^{n} - 1) / (q - 1)$	Both identities together, over a finite field

Frequently Asked Questions

Why is the projective space defined with a right action?

So that the scalar action and the coordinates sit on the same side. In the injectivity computation one arrives at $k = k' k''$ with $k''$ multiplying from the right; if $V$ were a left space acted on by right multiplication, the two sides would interfere and the coefficient comparison would fail. Over a commutative field the distinction evaporates, which is why it is rarely mentioned in classical projective geometry.

Does

(13.24)

mean

D^{*}

has no subgroups of finite index at all?

No. It says nothing about subgroups that are not of the form $K^{*}$ for a division subring $K$ . The theorem is specifically about division subrings, and the proof relies on $K$ being closed under addition — which an arbitrary subgroup of $D^{*}$ is not.

How does this compare with the situation for fields?

It is the same statement, and for fields it is classical: an infinite field $L$ with a proper subfield $K$ has $[L^{*} : K^{*}]$ infinite, because $ℙ^{1} (K)$ already has $| K | + 1$ points. The content of $(13.22)$ is that no commutativity is used anywhere, so the conclusion transfers to division rings without modification.

Why does

(13.26)

need Wedderburn's Little Theorem?

Because $(13.25)$ requires $D$ to be infinite before it can conclude that the class is infinite. Noncentrality of $a$ makes $D$ noncommutative, and it is Wedderburn's theorem that upgrades noncommutative to infinite. Without it, a hypothetical finite noncommutative division ring would have finite conjugacy classes and the argument would stall.

What is the exact size of a conjugacy class, not just its finiteness?

It is $[D^{*} : C_{D} (a)^{*}]$ , by orbit–stabiliser. Scott's theorem sharpens $(13.26)$ to say that for noncentral $a$ this index has the same cardinality as $D$ itself, so the class is as large as the ring allows.

Can

(13.23)

really be applied to sets that are not subspaces?

Yes — the proof only needs $1$ and one element $v_{2} \notin K$ inside $V$ , plus $v_{2} k \in V$ and $1 + v_{2} k \in V$ . Any subset closed under right multiplication by $K$ and under addition of elements of $K$ qualifies. This flexibility is what lets the corollary be applied to orbit sets that arise inside a larger ring without a natural module structure.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §13, (13.22)–(13.26), pp. 224–225.
C. Faith, “On conjugates in division rings”, Canadian Journal of Mathematics 10 (1958), 374–380.
I. N. Herstein, Noncommutative Rings, Carus Mathematical Monographs 15, Mathematical Association of America, 1968, Chapter 3.
P. Draxl, Skew Fields, London Mathematical Society Lecture Note Series 81, Cambridge University Press, 1983.
E. Artin, Geometric Algebra, Interscience, 1957, Chapter II (projective geometry over division rings).

AI Suggested Questions

Prove that the left and right projective spaces of a noncommutative division ring have the same cardinality but are not naturally isomorphic.
Give a proof of Scott's theorem that the conjugacy class of a noncentral element has cardinality equal to that of the division ring.
State Faith's theorem on the index of the normalizer $N_{D^{*}} (K^{*})$ and compare it with $(13.24)$ .
Which subgroups of finite index can $D^{*}$ have when $D$ is an infinite division ring?
Derive the point count of $ℙ^{n} (𝔽_{q})$ from the two counting identities on this page.
How does the identification $ℍ^{*} / ℂ^{*} ≅ ℙ^{1} (ℂ)$ relate to the Hopf fibration?
Does $(13.24)$ have an analogue for simple artinian rings and their unit groups?

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

The line, and why it is injective

Why dimension at least two

From vector spaces to arbitrary closed subsets

06Key Results

07Proof Techniques and Method

08Worked Example

ℂ inside ℍ: the index is the Riemann sphere

The same count as a conjugacy class

The degenerate finite case

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

Counting points of projective spaces

Projective codes

Subgroups of finite index

Honest summary

12Standards and Notation

13Failure Modes and Common Mistakes

14Quick Reference

15Frequently Asked Questions

16Related KEVOS Topics

17References

18AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Worked Example

$ℂ$ inside $ℍ$ : the index is the Riemann sphere

Comparison and Classification

Relationship Map

Applications and Industry Use

Standards and Notation

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions