0
votes
1answer
36 views

The anti-symmetrization and simetrization operators are mutually orthogonal

For each vector $x=(x_1,\dots,x_n)$ of an $n$-dimensional vector space $V$, and for each permutation $s$ of the symmetric group on the $n$-element set $S_n$, put $s(x)=(x_{s(1)},\dots,x_{s(n)})$. Then ...
3
votes
1answer
84 views

Conjugation on subgroups of $A_4$ faithful?

Let $X$ be the set of all subgroups of $G=A_4$. We define the group action $$G\times X\ni(g,H)\mapsto gHg^{-1}\in X$$ I am trying to determine whether this action is faithful, i.e. $\bigcap_{H\in X} ...
1
vote
2answers
93 views

Finding the number of orbits

How many orbits are there of $(12)(25)$ in $S_{5}$? Considering the permutation $(12)$, it has $4$ orbits and is as follows: $\{\{1,2\},\{3\},\{4\},\{5\}\}$ and (25) also has 4 orbits and is also ...
9
votes
4answers
174 views

Group Action - Permutation on the Polynomial

I'm trying to check the permutation on the polynomial is a Group Action, but I'm not getting the second axiom. I'm following my lecturer's work --- Examples 2.1 and 2.6 on page 5 on ...