I had the same question after the example of S_4 wreath S_3, where there are only 3 copies of S_4 but |S_3|=6. Looking on Wikipedia, it looks like the fuller definition is you have H acting on a finite set Omega, and then, for G wreath H, you take Omega copies of G and have H act on these via it's action on Omega. When Omega = H with the action of left multiplication this is called the "regular wreath product", like the Z_2 wreath Z_2 example --- but the S_4 wreath S_3 example is not a regular wreath product, it's implied there that Omega is a 3-element set with an obvious action by H=S_3.
Nice picture @7:40! Reminds me of the visualization of finite abelian groups as combination locks that I first saw in this video th-cam.com/video/Ct2fyigNgPY/w-d-xo.html&ab_channel=MathVisualized. I like when people give me geometric ways of picturing groups (too bad this sort of stuff doesn't appear in textbooks, or at least the textbooks I've read).
Very clear exposition of wreath products, I was struggling with those for a while, thank you!
Very nice ! By the way, the sign in the wreath product G \wr H is read "wreath".
Wonderful!!!
Thank you for the video - I ahve a question: when you say we take (G X G X ... X G) H times do you mean |H| times?
I have to think that is what he means, especially in light of the example he gives following that statement. I agree that "H times" is meaningless.
I had the same question after the example of S_4 wreath S_3, where there are only 3 copies of S_4 but |S_3|=6. Looking on Wikipedia, it looks like the fuller definition is you have H acting on a finite set Omega, and then, for G wreath H, you take Omega copies of G and have H act on these via it's action on Omega. When Omega = H with the action of left multiplication this is called the "regular wreath product", like the Z_2 wreath Z_2 example --- but the S_4 wreath S_3 example is not a regular wreath product, it's implied there that Omega is a 3-element set with an obvious action by H=S_3.
helpful video
Nice picture @7:40! Reminds me of the visualization of finite abelian groups as combination locks that I first saw in this video th-cam.com/video/Ct2fyigNgPY/w-d-xo.html&ab_channel=MathVisualized. I like when people give me geometric ways of picturing groups (too bad this sort of stuff doesn't appear in textbooks, or at least the textbooks I've read).