1

So this is a pretty general question, and I'm not even sure what it is I should be asking. Basically my abstract algebra textbook does not clearly define conjugations and just sort of throws out this:

$$g⋅a = gag^{-1}$$

I don't know what this comes from or what it is saying, but from my trying to figure this out, it seems like it comes up a lot, and I just can't make sense of it.

I keep trying to visualize everything group-related as symmetric permutations, but I imagine this is very limited, especially when considering a group acting on itself. (How does that work?) Are there other ways of visualizing what this is trying to say?

Let me know if providing the hw question that led to this confusion is worth commenting

Shaun
  • 44,997
Deck
  • 11
  • 2
    Conjugation is a bit like a translation. If you know about Cayley graphs, it basically changes your base point from the identity to $g$. It says "travel to $g$, do $a$, then travel back along $g$", and basically you translate $a$ by $g$. (This would be clearer with a picture, but then that's why it's just a comment :-) ) – user1729 Oct 26 '20 at 21:40
  • 1
    Elements $a,b$ in a group $G$ are called conjugate if there is some element $g\in G$ such that $b=gag^{-1}$. It is easy to check that this defines an equivalence relation on $G$. For example, in the group $GL_n(F)$, two matrices are conjugate if they are similar. – Mark Oct 26 '20 at 21:41
  • 1
    In linear algebra this is related to change of basis. So is like shifting an element to represent the same element but with different eyes.(the $g$ eyes). – Phicar Oct 26 '20 at 21:43
  • Which textbook are you referring to? – Shaun Oct 26 '20 at 21:46
  • NB: cf. inner automorphisms. – Shaun Oct 26 '20 at 21:51
  • I closed this as a duplicate of an older question, which has lots of good answers. If the answers there don't help then you should edit your question to say why they don't help. – user1729 Oct 27 '20 at 10:18

2 Answers2

0

I learned a slightlly diffrent definition but i'm pretty sure they are related. So essentialy the conjugations of two elements in a group measures how much they overlap, so for example in the symetric group any two elements that dont act on the same numbers would have conjugations e (The identity) which means they dont affect each other, we would write the conjugations of two elements as ga(g-1)*(a-1)

sean python
  • 132
0

The notation $g⋅a = gag^{-1}$ defines a group action of $G$ on itself, that is an application $\phi : G \times G \to G$ such that $\phi(g,a)=g⋅a = gag^{-1}$, $e⋅a=a$ and $g'⋅(g⋅a)=(g'g)⋅a$.

If you're familiar with linear algebra, you can think of $g⋅a$ as a change of basis. If you take $G = (\Bbb {GL}_n(K), \times)$ for example, you see that $g⋅a$ represents a change of basis for the matrix $a$.

Michelle
  • 1,784
  • 7
  • 20