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An ageing Algebra teacher


9h
answered Equation Theory: A polynomial with specific remainders when divided by specific divisors. What is the remainder when divided by BOTH divisors
14h
comment Elements whose orders are multiple of $p$
Note This is an answer to the original question (which was later edited), in which $N$ was cyclic, of order $p$.
1d
revised Something related to Frobenius coin Problem/Chicken McNugget Theorem
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1d
revised Pairs of $2\times 2$ matrices generating free groups.
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1d
revised Pairs of $2\times 2$ matrices generating free groups.
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1d
answered Pairs of $2\times 2$ matrices generating free groups.
2d
answered Elements whose orders are multiple of $p$
Jul
9
comment Calculating all potencies of a Matrix
[Follows from previous comment] I ask my students: would you prefer to have to compute the tenth power of $\begin{bmatrix}-1&0\\0&2\\\end{bmatrix}$ or $\begin{bmatrix}0&2\\1&1\\\end{bmatrix}$? Since the second matrix conjugates to the first one, it is nearly the same.
Jul
9
comment Calculating all potencies of a Matrix
Take any matrix $B$. Suppose you have to compute a high power $B^{n}$ of $B$. IF all of the eigenvalues of $B$ are in the underlying field, you can conjugate $B$ to a matrix that consists of diagonal blocks of a form similar to $a E + A$ (of various dimensions), where the $a$ are the eigenvalues. The exewrcise shows that computing powers of these $A$ is easy. So you compute those powers, and then conjugate back to get $B^{n}$. I use a simplified example of this kind to justify the introduction of eigenvalues and eigenvectors, as in the next comment [follows in the next comment]
Jul
9
revised Calculating all potencies of a Matrix
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Jul
9
answered When does the system of equations have initly solution, no solutions, and only one solution
Jul
9
answered Calculating all potencies of a Matrix
Jul
9
comment Order of the elements of a right coset
In general, very little. Think of $A_{n} \le S_{n}$, where each of the two cosets of $A_{n}$ contain elements of many different orders. There are some particular cases in which one can say something. For instance if $G$ is a finite nilpotent group which is metabelian (i.e., the derived subgroup $G'$ is abelian) and can be generated by two elements, then all the elements in a coset $a G' \ne G'$ have the same order.
Jul
8
comment Proving the Thompson Transfer Lemma
You have written $M$ an index $2$ subgroup of $M$. Possibly the second $M$ is a $T$?
Jul
8
revised Help with understanding definition of divisibility in this case.
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Jul
8
revised Help with understanding definition of divisibility in this case.
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Jul
8
answered Help with understanding definition of divisibility in this case.
Jul
8
revised Example of a bijection from the set of real numbers to a subset of irrationals
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Jun
30
answered Similar matrices that are not diagonalizable
Jun
29
comment S+N decomposition
Look, it's simple, $A = - I + (A + I)$, with $-I$ semisimple and $A + I$ nilpotent, so that's it!