inner automorphism
Technical / Very LowFormal, Technical, Academic (specifically mathematical)
Definition
Meaning
In group theory, an automorphism of a group defined by conjugation by a fixed element of the group.
A function from a group to itself that is both an isomorphism and is defined by the operation of taking an element, multiplying by a specific fixed element on one side, and then by its inverse on the other side. It is an automorphism that is essentially a relabeling of the group's elements that respects its structure and arises intrinsically from the group itself.
Linguistics
Semantic Notes
The concept is exclusively mathematical and part of abstract algebra. It is a subtype of 'automorphism', distinguished by being defined via conjugation. The set of all inner automorphisms forms a normal subgroup of the full automorphism group.
Dialectal Variation
British vs American Usage
Differences
No lexical or definitional differences. Spelling conventions follow respective norms for surrounding text (e.g., 'centre' vs. 'center' in definitions).
Connotations
None beyond the strict mathematical meaning.
Frequency
Identically rare and confined to advanced mathematics contexts in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
The inner automorphism [induced by/defined by/conjugation by] gAn inner automorphism of [group name]The map φ_g is an inner automorphism.Vocabulary
Synonyms
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Never used.
Academic
Exclusively used in advanced mathematics, particularly in abstract algebra, group theory, and related research papers and textbooks.
Everyday
Never used.
Technical
The primary context. Used with precise definition in mathematics lectures, seminars, and technical discussions among mathematicians.
Examples
By Part of Speech
verb
British English
- One can inner-automorph a group element by conjugating it.
- The operation inner-automorphises the entire structure.
American English
- One can inner-automorph a group element by conjugating it.
- The operation inner-automorphizes the entire structure.
adverb
British English
- The map acts inner-automorphically on the subgroup.
- The transformation was applied inner-automorphically.
American English
- The map acts inner-automorphically on the subgroup.
- The transformation was applied inner-automorphically.
adjective
British English
- The inner-automorphism group is normal.
- We studied the inner-automorphism structure.
American English
- The inner-automorphism group is normal.
- We studied the inner-automorphism structure.
Examples
By CEFR Level
- This word is not used at the A2 level.
- This word is not used at the B1 level.
- In mathematics, an 'inner automorphism' is a special kind of symmetry for a set with operations.
- The proof relied on showing that the proposed map was not merely an automorphism, but specifically an inner automorphism induced by a central element.
Learning
Memory Aids
Mnemonic
INNER automorphisms come from INside the group, via INternal conjugation (IN by g and INverse of g).
Conceptual Metaphor
A SYMMETRY FROM WITHIN: Viewing the group's structure from a different perspective, but a perspective that is already available inside the group itself, like rotating a symmetrical object around its own axis.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Прямой перевод «внутренний автоморфизм» является точным и стандартным. Ловушок в переводе нет, но важно понимать точное математическое определение.
Common Mistakes
- Confusing 'inner automorphism' with a general automorphism.
- Forgetting that the defining element must be from the group itself.
- Incorrectly stating that all automorphisms are inner (e.g., for abelian groups, only the identity is inner).
- Misspelling 'automorphism'.
Practice
Quiz
What distinguishes an inner automorphism from an outer automorphism?
FAQ
Frequently Asked Questions
No. An automorphism is inner only if it can be expressed as conjugation by some group element. Many groups have automorphisms that are not of this form, called outer automorphisms.
The kernel is the center of the group, Z(G), consisting of elements that commute with all other elements.
No. In an abelian group, conjugation by any element g is the identity map (gxg⁻¹ = x), so the only inner automorphism is the trivial one.
They are important because they are automorphisms that are intrinsically defined by the group's own structure. The set of all inner automorphisms forms a normal subgroup of the full automorphism group, and the quotient is the outer automorphism group, which is a key invariant in group theory.