Questions tagged [group-theory]

The study of algebraic structures known as groups.

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A basic mistake by Cayley

Arthur Cayley's first paper on abstract groups, in 1854, can be found in his Collected Papers on the Internet Archive, starting at https://archive.org/stream/collectedmathema02cayluoft#page/122/mode/...
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369 views

Whence “homomorphism”, “homomorphic”?

The kernel question leads to another : Today, homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen? “Homomorphic” ...
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187 views

Reflections in the 18th century

It is well known that the theory of reflections was considerably developed during the 19th century with the development of group theory (e.g. Klein) and the theory of transformations. However, I'm ...
4
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94 views

Lagrange's Theorem as he stated it

In Wikipedia, I found that Lagrange did not state Lagrange's Theorem in its general form. He stated "If a Polynomial in $n$ variables has its variables permuted in all $n!$ ways, the number of ...
3
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1answer
113 views

Origin of the Recurrence Relation for Clebsch-Gordan Coefficients

The Clebsch-Gordan coefficients $C_{\pm }(J,M)$ arise in quantum mechanics in the problem of addition of angular momentum. They also arise in mathematics in the more theoretical (but related) problem ...
3
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0answers
90 views

Origin of the term 'index of a subgroup'

The index of a subgroup $H$ in a group $G$ is the number of distinct cosets of $H$ in $G$. Why did someone decide to call this an 'index'?
3
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0answers
75 views

First uses of the Ping-Pong Lemma

I am interested in knowing the origins of this useful result, but I haven't been able to precisely pinpoint the context of its first use. Most texts seem to indicate the result originally comes from ...
3
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0answers
182 views

Writing functions on the right

In group theory, writing functions on the right is a common, though not universal practice. Thus, given mappings $f$, $g$ and group element $\alpha$, one might write $\alpha f$ and $\alpha (f \circ ...
3
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0answers
177 views

History of the Wreath product

Why is the wreath product so named? If possible, please provide a citation.
2
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0answers
62 views

Continuation on Galois’ lost memoir(s): was Poisson right in his review of Galois’ memoir?

A recent question asked whether Galois’ “lost” memoir has been since recovered (similar to Abel’s lost memoir). The memoir referenced in the question is the one Galois submitted to the French “...
2
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0answers
93 views

Why are faithful actions called faithful and who first called them faithful?

This is a cross post from MSE I want to know why are faithful actions called faithful and who first called them faithful? Definition: An action $G$ on $X$ is faithful when ${g_1 \neq g_2 \Rightarrow ...
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65 views

Any idea on how Lagrange came up with similar functions concept in (proto)group theory?

Lagrange defines "similar functions" as functions of the roots of an equation where they change values only at the same kind of permutations of the roots. What's a possible predecessor of the idea of ...
2
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371 views

What does the "G" for the similitude groups stand for?

When we have a bilinear symmetric/ bilinear anti-symmetric/hermitian form $b$ on a real/complex vector space $V$, one can consider the group of invertible matrices $A \in GL(V)$ which respect $b$, ...