3
votes
2answers
42 views

Working with groups. Finding the inverse of some $S_9$

I want to compute the inverse of: $\begin{pmatrix} 1&2&3&4&5&6&7&8&9\\3&2&1&6&5&9&4&8&7 \end{pmatrix}$ Sorry about alignment(they are ...
0
votes
3answers
68 views

If Q is a p-Sylow-Group of H there is a p-Sylow-Group P of G with $\phi(P)=Q$ while $\phi:G\rightarrow H$ epimorphism

Let G be a finite group and $\phi: G \rightarrow H$ a group-epimorphism. Proof: If $Q\in Syl_p(H)$ there is a $P\in Syl_p(G)$ with $Q=\phi(P)$.
0
votes
1answer
70 views

Find the inverse of a matrix in $GL(2\,,\, \Bbb Z_{11})$.

What are the necessary steps and reasoning for calculating the following matrix in GL(2,$\Bbb Z_{11}$): $M = \begin{pmatrix} 2&6 \\3&5 \end{pmatrix}$. I found the answer to be ...
0
votes
2answers
42 views

Uniqueness of Inverses in Groups Implies Associativity Holds?

I was checking multiplication tables for groups with 4 elements, to see which tables "passed" the group axioms of closure, associativity, identity and inverses. But then I had a question, so hopefully ...
2
votes
1answer
64 views

Monoid with inversion

Is there a name for monoid with operation $a\mapsto a^{-1}$ conforming the equations $(a^{-1})^{-1}=a$ and $(b\cdot a)^{-1} = a^{-1}\cdot b^{-1}$? (with no requirement that $a^{-1}\cdot a$ would be ...
0
votes
2answers
245 views

Rings | Homomorphisms | Units

Question Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$. Attempt ...
2
votes
3answers
96 views

The inverse of the inverse in a group

In one of my math exercises, I'm being asked to prove that for all $a, b \in G | (a^{-1})^{-1} = a$ with G a group. However, nowhere is stated that it is a commutative group. My first thought was ...
4
votes
2answers
182 views

Is $a^{-1} + b^{-1} = (a + b)^{-1}$ always true for Abelian group?

I get the equation $a^{-1} + b^{-1} = (a + b)^{-1}$ from ordinary + operation. For ordinary + operation I mean $a^{-1} = -a$. It is also true for * of rational numbers $3^{-1}*4^{-1} = \frac{1}{3} * ...
1
vote
1answer
161 views

Terminology question; inverse vs complement in Boolean algebra

This was said at a lecture I attended: $e$ is neutral element for operation $*$ if $\forall x (x*e=x \wedge e*x = x)$. So, for example 0 is n. e. for disjunction and 1 is n. e. for ...
2
votes
3answers
132 views

Determine invertible and inverses in $(\mathbb Z_8, \ast)$

Let $\ast$ be defined in $\mathbb Z_8$ as follows: $$\begin{aligned} a \ast b = a +b+2ab\end{aligned}$$ Determine all the invertible elements in $(\mathbb Z_8, \ast)$ and determine, if possibile, ...
2
votes
2answers
146 views

15Puzzle, sum of inversions - what's been summed?

According to this page: http://mathworld.wolfram.com/15Puzzle.html it says that While odd permutations of the puzzle are impossible to solve I've red this article: ...