a fundamental group

USAGE SUMMARY

The phrase "a fundamental group" is correct and usable in written English.
It can be used in mathematical contexts, particularly in topology, to refer to a specific algebraic structure associated with a topological space. Example: "In algebraic topology, the fundamental group provides important information about the shape and structure of a space."

✓ Grammatically correct

Science

Encyclopedias

Human-verified examples from authoritative sources

Exact Expressions

2 human-written examples

Gerhardt suggested that successive additions of methylene (CH2) to a fundamental group yielded a series of related compounds, such as the alcohols.

Encyclopedia Britannica

Cluster discovery techniques are a fundamental group of exploratory methods designed to identify areas exhibiting elevated levels of disease, risk, danger, etc.

Spatial Statistics

Human-verified similar examples from authoritative sources

Similar Expressions

58 human-written examples

Kong, A digital fundamental group, Comput.

Information Sciences

Many of the restrictions for a group (Gamma ) to be a projective group (fundamental group of a projective variety X) are indeed restrictions to be a Kähler group.

Bulletin of Mathematical Sciences

These manifolds are simply connected, hence by van Kampen we introduce the relation ( R_j x) = 1), and finally one obtains a M with fundamental group (cong Gamma ).

Bulletin of Mathematical Sciences

In the case where the action of G is no longer free, we would like to remember the group (Gamma ), in view of its importance, even if it is no longer a 'bona fide' fundamental group.

Bulletin of Mathematical Sciences

The Schur property (SP) plays an important role in order to show that an Abelian variety with such a G-action on its fundamental group must split as a product.

Bulletin of Mathematical Sciences

We show that any countable subgroup of the multiplicative group R×+ of positive real numbers can be realized as the fundamental group F(A) of a separable simple unital C⁎-algebra A with unique trace.

Journal of Functional Analysis

However, for a closed surface that allows a hyperbolic metric and is equipped with a canonical set of fundamental group generators, the shortest word problem can be reduced to finding the shortest loop that is homotopic to the given loop, which can be solved efficiently.

Computer-Aided Design

(Gromov's few relations theorem) Let X be a compact Kähler manifold and assume that there exists a surjection of its fundamental group begin{aligned} Gamma : = pi _1 (X) rightarrow G = langle x_1, ldots, x_n | R_1 x), ldots, R_m (x) rangle, end{aligned}onto a finitely presented group that has 'few relations', more precisely where ( n ge m-2).

Bulletin of Mathematical Sciences

It depends on a representation of the fundamental group.

Journal of Functional Analysis

Expert writing Tips

Best practice

When writing about mathematical or topological concepts, ensure that you define "a fundamental group" clearly, as its meaning is specific to these fields. Use precise language to avoid ambiguity.

Common error

Do not use "a fundamental group" in contexts where you simply mean a basic or essential group. The term has a specific mathematical meaning, and using it loosely can lead to confusion. Instead use alternatives like "core group" or "basic group" in general contexts.

Linguistic Context

The phrase "a fundamental group" functions primarily as a noun phrase. It identifies a specific algebraic structure used in mathematics, particularly in topology. As Ludwig AI explained, the phrase is correct and mostly found in mathematical contexts.

Expression frequency: Rare

Frequent in

Science

60%

Encyclopedias

20%

Wiki

20%

Less common in

News & Media

0%

Formal & Business

0%

Social Media

0%

Ludwig's WRAP-UP

In summary, the phrase "a fundamental group" is a technical term primarily used within the fields of mathematics and topology. As confirmed by Ludwig AI, it is grammatically correct and serves to denote a specific algebraic structure. Its usage is largely confined to scientific and academic contexts, making it unsuitable for informal communication. When writing, ensure clarity and precision, as the term carries a specific mathematical meaning. If referring to a basic group in a non-mathematical setting, opt for alternative phrases like "core group" or "essential group" to avoid confusion.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

foundational group

Replaces "fundamental" with "foundational", highlighting the group's role as a basis for something else.

essential group

Uses "essential" instead of "fundamental", emphasizing the group's indispensability.

key group

Replaces "fundamental" with "key", emphasizing the group's critical or vital role.

basic group

Replaces "fundamental" with "basic", indicating a core or essential group.

principal group

Substitutes "fundamental" with "principal", highlighting the group's leading or dominant role.

primary group

Employs "primary" in place of "fundamental", denoting the main or principal group.

central group

Uses "central" instead of "fundamental", indicating the group's importance or position.

core group

Substitutes "fundamental" with "core", emphasizing the central or most important group.

underlying group

Replaces "fundamental" with "underlying", suggesting a group that forms the basis of a structure.

base group

Uses "base" instead of "fundamental", suggesting a foundational or underlying group.

FAQs

What is "a fundamental group" in mathematics?

In mathematics, specifically algebraic topology, "a fundamental group" is an algebraic group that captures basic information about the shape of a topological space. It describes the loops in the space and how they can be deformed into each other.

How do I use "a fundamental group" in a sentence?

When discussing topological spaces, you might say, "The sphere has a trivial fundamental group", indicating that any loop on the sphere can be continuously shrunk to a point.

What are some simpler alternatives to "a fundamental group" in non-mathematical contexts?

In contexts outside of mathematics, you can use alternatives like "core group", "basic group", or "essential group" to convey the idea of a central or foundational group without the technical meaning.

Which is the correct term, "fundamental group" or "the fundamental group"?

Both "fundamental group" and "the fundamental group" are correct, but "the fundamental group" is generally used when referring to a specific group associated with a particular topological space. "A fundamental group" is more general, discussing the concept.