jordan-holder theorem
Very lowTechnical (mathematics)
Definition
Meaning
A fundamental theorem in group theory stating that any two composition series of a finite group are equivalent.
The theorem guarantees a notion of uniqueness for the simple group factors (composition factors) appearing in a composition series, analogous to the uniqueness of prime factors in integer factorization. It is a central result in the structure theory of finite groups.
Linguistics
Semantic Notes
This is a proper noun referring to a specific, named theorem in abstract algebra. It is always capitalized and typically hyphenated (Jordan-Holder).
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or spelling. Pronunciation of the individual names 'Jordan' and 'Holder' follows general British/American patterns.
Connotations
Purely technical, with no regional connotative differences.
Frequency
Used exclusively in advanced mathematical discourse with equal rarity in both varieties.
Vocabulary
Collocations
Grammar
Valency Patterns
The [NOUN: theorem/result] [VERB: states/implies/guarantees] that...Vocabulary
Synonyms
Strong
Neutral
Weak
Usage
Context Usage
Business
Never used.
Academic
Exclusively used in advanced mathematics lectures, textbooks, and research papers, particularly in algebra and group theory.
Everyday
Never used.
Technical
Core terminology in abstract algebra and finite group theory.
Examples
By CEFR Level
- The Jordan-Holder theorem is a very advanced maths idea.
- In university maths, students learn about the Jordan-Holder theorem in group theory.
- The Jordan-Holder theorem ensures that the simple factors of a group's composition series are uniquely determined.
- By invoking the Jordan-Holder theorem, one can demonstrate that the composition factors of a finite solvable group are cyclic of prime order.
Learning
Memory Aids
Mnemonic
Imagine a JORDANian king (Jordan) HOLDing aER (Holder) a unique set of building blocks (simple groups). Just as his building block tower can be rebuilt in different orders but must use the same blocks, the theorem says composition series have the same factors.
Conceptual Metaphor
UNIQUE FACTORIZATION / BUILDING BLOCKS (The theorem treats groups as being built from unique, indivisible 'blocks' - the simple composition factors).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- May be transliterated directly as 'теорема Жордана-Гёльдера' (with umlaut in Гёльдер). Ensure both names are included and capitalized.
- Do not confuse with 'Jordan curve theorem' (теорема Жордана о кривой), which is a different result in topology.
Common Mistakes
- Misspelling as 'Jordan Holder theorem' (without hyphen).
- Incorrect pronunciation of 'Holder' as /ˈhɒldə/ instead of /ˈhəʊldə/ (UK) or /ˈhoʊldər/ (US), which anglicizes the original German name 'Hölder'.
- Using it to refer to infinite groups without the proper generalization (the theorem, in its basic form, is for finite groups).
Practice
Quiz
What is the primary domain of the Jordan-Holder theorem?
FAQ
Frequently Asked Questions
In its classic form, yes, it applies to finite groups. There are generalizations for modules and certain infinite groups with composition series.
Camille Jordan (French) and Otto Hölder (German) were mathematicians who independently contributed to the development and proof of this theorem in the late 19th century.
A composition series is a finite sequence of subgroups, each normal in the next, with the quotients (called composition factors) being simple groups.
It provides a rigorous foundation for analyzing the structure of groups by breaking them down into unique, fundamental components (simple groups), much like prime factorization for integers.