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3

Try these excellent books: An Introduction to the Theory of Groups by Rotman. The Theory of Groups by Marshall Hall Jr.


3

If $G$ is any non-trivial group, then $\mathrm{Aut}(G \times G)$ is not abelian. This is an improvement of Hagen von Eitzen's observation. Proof. If $G$ is abelian, then $(x,y) \mapsto (y,x)$ does not commute with $(x,y) \mapsto (x,x+ y)$. If $G$ is not abelian, we find a nontrivial inner automorphism $\sigma$ of $A$. But then $(x,y) \mapsto (y,x)$ ...


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In general, given a finite group $G$, you can find its conjugacy classes by choosing an element $x$, and computing $g^{-1}xg$ for all $g$ in $G$; this will give you the conjugacy class of $x$. Then pick an element $y$ which isn't in the class of $x$, and compute $g^{-1}yg$ for all $g$ in $G$; this will give you the conjugacy class of $y$. If $G$ has any ...


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Observe first that $$ x_2x_1=x_1^2x_2,\quad x_3x_2=x_2^2x_3,\quad x_1x_3=x_3^2x_1 $$ and more generally $$ x_2^jx_1^k = x_1^{2^jk} x_2^j,\qquad x_3^jx_2^k = x_2^{2^jk} x_3^j,\qquad x_1^jx_3^k = x_3^{2^jk} x_1^j $$ for any $j,k\in\mathbb{N}$. Then $$ x_1^2 x_2^2 x_3 \;=\; x_1^2 x_3 x_2 \;=\; x_3^4 x_1^2 x_2 \;=\; x_3^4 x_2 x_1 \;=\; x_2^{16}x_3^4x_1 \;=\; ...


1

Major hint: Say $\phi_1,\phi_2\in S_5$ are conjugate by $\psi$ (that is, $\phi_2=\psi\phi_1\psi^{-1}$). Say the representation of $\phi_1$ as a product of disjoint cycles is $$\phi_1=(a,b,c)(d,e).$$Show that $$\phi_2=(\psi(a),\psi(b),\psi(c))(\psi(d),\psi(e)).$$ Looks like one conjugacy class in $S_n$ for each way of writing $n$ as a sum of positive ...


1

Looks good to me. As a further check you can do the following. The trace of a 2D rotation matrix is $2\cos\theta$ where $\theta$ is the angle of rotation, because its eigenvalues are $e^{\pm i\theta}$ (if you next need the trace of a 3D rotation, then you need to add $+1$ for the eigenvalue $+1$ from the axis of rotation). The trace of an nD reflection is ...


1

If $G=\mathbb Z[\frac12]$, then $G$ is torsion-free and $\{1\}$ is a maximal rationally independent set. Furthermore, $G$ is not finitely generated as you already know from this answer. It's not hard to prove that $G$ is not a $\mathbb Q$-vector space: since $(1,x)\mapsto x$ for all $x\in G$ then we should have $(q,x)\mapsto qx$ for all $q\in\mathbb Q$ and ...


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Martin Isaac's book Algebra might be useful to you. It is at the graduate level so it might be useful if you are going through the material a second time.


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Although I believe that Dummit and Foote's Abstract Algebra combined with Herstein's Topics in Algebra which has excellent exercises is the perfect recipe for the job, I would recommend two more books for the study of finite groups. The first one is Daniel Gorenstein's Finite Groups which is a book of great depth and covers a lot of material about groups. ...


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I enjoyed Michael Artin's book Algebra. This would still be as you say the "group theory" part of an abstract algebra book, but it is rigorous and touches on the topics you have listed.


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This is true for any semigroup (even if it is not a monoid) and is easy to prove if you know about Green's relations. If $S$ be a semigroup and $e$ is an idempotent of $S$, then $eSe$ is a monoid (with identity $e = eee$). Suppose that $e$ and $f$ are idempotents such that $eSe$ and $fSf$ are groups. Then in particular, $e \mathrel{\mathcal H} efe$ and $f ...



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