science

Three subgroups lemma

Three subgroups lemma is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Three subgroups lemma rather than just read about it. In short: In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity.

Key takeaways

Three subgroups lemma belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Three subgroups lemma to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Three subgroups lemma from memory before moving on to harder problems.

Reference excerpt

In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity.

Notation In what follows, the following notation will be employed:

If H and K are subgroups of a group G, the commutator of H and K, denoted by [H, K], is defined as the subgroup of G generated by commutators between elements in the two subgroups. If L is a third subgroup, the convention that [H,K,L] = [[H,K],L] will be followed. If x and y are elements of a group G, the conjugate of x by y will be denoted by x y {\displaystyle x^{y}} . If H is a subgroup of a group G, then the centralizer of H in G will be denoted by CG(H).

Statement Let X, Y and Z be subgroups of a group G, and assume

[ X , Y , Z ] = 1 {\displaystyle [X,Y,Z]=1} and [ Y , Z , X ] = 1. {\displaystyle [Y,Z,X]=1.}

Then [ Z , X , Y ] = 1 {\displaystyle [Z,X,Y]=1} . More generally, for a normal subgroup N {\displaystyle N} of G {\displaystyle G} , if [ X , Y , Z ] ⊆ N {\displaystyle [X,Y,Z]\subseteq N} and [ Y , Z , X ] ⊆ N {\displaystyle [Y,Z,X]\subseteq N} , then [ Z , X , Y ] ⊆ N {\displaystyle [Z,X,Y]\subseteq N} .

Proof and the Hall–Witt identity Hall–Witt identity If x , y , z ∈ G {\displaystyle x,y,z\in G} , then

[ x , y − 1 , z ] y ⋅ [ y , z − 1 , x ] z ⋅ [ z , x − 1 , y ] x = 1. {\displaystyle [x,y^{-1},z]^{y}\cdot [y,z^{-1},x]^{z}\cdot [z,x^{-1},y]^{x}=1.}

Proof of the three subgroups lemma Let x ∈ X {\displaystyle x\in X} , y ∈ Y {\displaystyle y\in Y} , and z ∈ Z {\displaystyle z\in Z} . Then [ x , y − 1 , z ] = 1 = [ y , z − 1 , x ] {\displaystyle [x,y^{-1},z]=1=[y,z^{-1},x]} , and by the Hall–Witt identity above, it follows that [ z , x − 1 , y ] x = 1 {\displaystyle [z,x^{-1},y]^{x}=1} and so [ z , x − 1 , y ] = 1 {\displaystyle [z,x^{-1},y]=1} . Therefore, [ z , x − 1 ] ∈ C G ( Y ) {\displaystyle [z,x^{-1}]\in \mathbf {C} _{G}(Y)} for all z ∈ Z {\displaystyle z\in Z} and x ∈ X {\displaystyle x\in X} . Since these elements generate [ Z , X ] {\displaystyle [Z,X]} , we conclude that [ Z , X ] ⊆ C G ( Y ) {\displaystyle [Z,X]\subseteq \mathbf {C} _{G}(Y)} and hence [ Z , X , Y ] = 1 {\displaystyle [Z,X,Y]=1} .

See also Commutator Lower central series Grün's lemma Jacobi identity

Notes

References I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.

Worked examples

Example 1 — a first encounter with Three subgroups lemma

Start with the simplest possible case. Write down what Three subgroups lemma claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Three subgroups lemma before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Three subgroups lemma ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Three subgroups lemma

In research: Three subgroups lemma appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Three subgroups lemma in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Three subgroups lemma is common in secondary-school and first-year university syllabi. It links to neighbouring topics Lemmas in group theory, so understanding it makes those chapters shorter.
In everyday life: Look for Three subgroups lemma outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Three subgroups lemma in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Three subgroups lemma means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Three subgroups lemma out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Three subgroups lemma in simple terms?

In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity.

Why does Three subgroups lemma matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Three subgroups lemma?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Three subgroups lemma.