# Tagged Questions

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### The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
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### References on the theory of $2$-groups.

Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
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### Details about “fingerprinting” algorithms for groups?

where can I find details about "Fingerprinting" algorithms (to test whether two groups are non-isomorphic) "‘Fingerprinting’: For every group $G_1,…, G_r$ evaluate various isomorphism-invariant ...
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### Sequence and series problems with slick group theoretic solutions.

So I'm trying to get this unmotivated student in my class interested in learning group theory. I've recently ascertained that he likes analysis a bit better than algebra, and that sequences and ...
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### Choosing an advanced group theory text: concerns

In this question, An Introduction to the Theory of Groups by Rotman is recommended twice as a good second-course group theory text. However, after reading the reviews here, and seeing this pdf of ...
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### Induced representations and degrees

Let $G$ be a finite group, $S$ a subgroup, $K$ an arbitrary (not necessarily algebraically closed) field whose characteristic does not divide the group order. 1) Let $\operatorname{Ind}_S^G(f)$ have ...
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### Burnside's Action Equivalence Criterion

Theorem: Actions of a finite group G on finite sets $X$ and $Y$ are equivalent if and only if $|X^H|=|Y^H|$ for each subgroup $H$ of $G$ I've proved the finite case inductively but I'm wondering ...
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### References about finite group theory

In your opinion which are the best books regarding the theory of finite groups? I think that a wonderful one is "Finite Group Theory - Michael Aschbacher". Many thanks.
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### Infinite index subgroups of $SL(3,\mathbf{Z})$

Finite index subgroups of $SL(3,\mathbf{Z})$ are well-know (at least we know that they are congruence subgroups). But I wasn't able to find reference on infinite index subgroups. Does someone knows ...
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### Involutions and Abelian Groups, II.

In the thread Involutions and Abelian Groups, I supplied a solution to the following interesting problem (with the help of hints provided by the OP). Let $G$ be a finite group and $I(G)$ the ...
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### Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
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### Is there a counterexample to this weakened converse of Hall's theorem?

Suppose that a finite group $G$ contains a Hall $\{p,q\}$-subgroup for every pair of prime divisors $p,q$ of $|G|$. Does it follow that $G$ is solvable?
For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let ...