Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,823
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Surjective maps given by words, redux
I asked some time ago:
Let $w(X,Y)$ be a word in $X$ and $Y$ (i.e., an element in the free group on $X$ and $Y$). Let the variables $x$ and $y$ now range among elements of $SL_n(K)$, $K$ an ...
- 19.1k
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3
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Computing a Factor Group
I have a problem in computing (i.e. classify) a factor group.
For example The group Z*Z*Z/<(3,6,9)> is isomorphic to Z_3*Z*Z. I can show this by contructing a homomorphism f
f(a,b,c) = ( a mod 3 ,...
- 567
5
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1
answer
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is amalgamation of groups associative
Given groups $G_1, G_2, G_3$ and injections $A_1 \to G_1$ and $A_1
\to G_2$ , from $A_2 \to G_2$ and $A_2 \to G_3$, let $G_1 *_{A_1} *G_2 *_{A_2} G_3$ be the amalgam formed these groups and maps.
...
- 515
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2
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Orbits of real groups, canonical forms of matrices
There are a lot of results in textbooks concerned with canonical forms of matrices under certain complex groups of transformations, e.g. GL(n|C), O(n|C),...
Could anybody give me references where the ...
- 93
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3
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Unipotent linear algebraic groups
Let $U_1$ be a unipotent group inside some Chevalley group $G$. For now, think of $G$ as being $SL_n(K)$ where $K$ is a field; then we can take $U_1$ to be a bunch of strictly upper triangular matrics....
- 11.1k
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4
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Is there a "universal group object"? (answered: yes!)
I want to say that a group object in a category (e.g. a discrete group, topological group, algebraic group...) is the image under a product-preserving functor of the "group object diagram", $D$. One ...
- 10.9k
12
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4
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Mystery of the Monstrous Moonshine
There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it,
...
- 14.8k
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Is ${\rm S}_6$ the automorphism group of a group?
The automorphism group of the symmetric group $S_n$ is $S_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S_6$ with the (cyclic) group of order $2$. ...
- 1,049
49
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1
answer
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Order of an automorphism of a finite group
Let G be a finite group of order n. Must every automorphism of G have order less than n?
(David Speyer: I got this question from you long ago, but I don't know whether you knew the answer. I stil ...
- 24.7k
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decidability of group homomorphism existence
Fix a finitely-presented group $G$ with distinguished non-identity element $g$. For any finitely-presented group $H$ with element $h$, is it decidable whether there is a homomorphism $h: G \...
- 87
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3
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Conjugation in SU(2)
For any two matrices $P,Q \in SU(2)$, with $tr(P)=tr(Q)=0$, does there always exist some $G\in SU(2)$ such that $G P G^{-1} = -P$, and $G Q G^{-1} = -Q\ ?$
- 1,109
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3
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Is any representation of a finite group defined over the algebraic integers?
Apologies in advance if this is obvious.
- 114k
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When is a map given by a word surjective?
Let $w(x,y)$ be a group word in $x$ and $y$.
Let $x$ and $y$ now vary in $\operatorname{SL}_n(K)$, where $K$ is a field. (Assume, if you wish, that $K$ is an algebraically complete field of ...
- 19.1k
5
votes
3
answers
889
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Two finite groups with the same identical relations?
An identical relation on a group G is a word w in Fr, the free group on r elements (for some r), such that evaluating w on any r-tuple of elements of G yields the identity (this just means ...
- 6,414
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votes
5
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959
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Non-conjugate words with the same trace
Let n>=2, p a large prime, G = SL_n(Z/pZ).
If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...
- 19.1k
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5
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Why are subfactors interesting?
I get asked this question a lot, and am not very happy with any of the answers.
Vaguely I think of subfactor theory as a generalization of representation theory of groups. That is, if you have a ...
- 241
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votes
5
answers
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How small can a group with an n-dimensional irreducible complex representation be?
More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex ...
- 114k
21
votes
5
answers
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Finite groups with the same character table
Say I have two finite groups $G$ and $H$ which aren't isomorphic but have the same character table (for example, the quaternion group and the symmetries of the square). Does this mean that the ...
- 9,982
12
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4
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Elements of infinite order in a profinite group
Say G is a profinite group with elements of arbitrarily large order. Do elements of infinite order exist (A) if we assume G is abelian? (B) in general?
A start for (A): we can ask the same question ...
- 10.9k
5
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1
answer
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When does a transitive action of a profinite group have an infinite orbit?
That is: suppose G is a profinite group acting 1-transitively (but maybe not regularly) on a set X. Is there a reasonable criterion for when there is a g in G and a point a in X such that the g-orbit ...
- 3,928
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When and how is a group of order n isomorphic to a regular subgroup of equal order?
In "Group Theory and Its Application to Physical Problems" by Morton Hamermesh, Morton states Cayley's theorem: Every group G of order n is isomorphic with a subgroup of the symmetric group Sn, which ...
18
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2
answers
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Homomorphism more than 3/4 the inverse
Suppose $G$ is a finite group and $f$ is an automorphism of $G$. If $f(x)=x^{-1}$ for more than $\frac{3}{4}$ of the elements of $G$, does it follow that $f(x)=x^{-1}$ for all $x$ in $G\ ?$
I know ...
- 5,207
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Is a quotient of a reductive group reductive?
Is a quotient of a reductive group reductive?
Edit [Pete L. Clark]: As Minhyong Kim points out below, a more precise statement of the question is:
Is the quotient of a reductive linear group by a ...
- 10.3k