Good video. The Klein 4 group can be generated by the set of 180-degree rotations about the axes x,y, and z in three dimensional space. While the Klein 4 group is finite and commutative, the group of rotations is infinite and non-commutative. Is the group generated by the standard quaternion basis the same as the group generated by 90-degree rotations about the x, y, and z axes?
of the klein 4? no it's not, remember H is a proper subgroup of G if H is a subgroup and H != G. So is a subgroup of the klein 4, but it's not the whole group ,so it's proper
@@TheMathSorcerer as far as I know If H is a subgroup of G and H=G or H= singleton set of identity element, then it's improper subgroup ... And here (e) is Identity ,so it's improper subgroup. Plz correct me if i am wrong..
Some authors of books use that as the definition. Depends on the books . I think it's more common to exclude the set with just the identity as this way the definition agrees with the set definition of a proper subset.
The German word "vier" (meaning 4) rhymes with the English "deer". Just put an "f" infront. Very similar words are also: - fear (tone down the a) - fierce (stop before the z loud)
When you write out a group's Caley table, you'll start to see that it resembles a game of Sudoku |e a b c -------------- e| e a b c a| a e c b b| b c e a c| c b a e See how in each row and each column there are no duplicate elements and every row has to have an e So it's interesting to work out all possible groups in an abstract sense You could use 0,1,2,3 instead of e,a,b,and c of course. Also the left most column and top most row always have to be the same for the identity element property to hold, so in this case there are 9 squares to play with
+The Math Sorcerer I really don't know why there are only 17 likes, this video is amazing, and all the work you did on Abstract Algebra is enormous and extremely valuable. Thank you very much! I have an exam tomorrow and I am revising using your videos. Thank you again!
Is there an infinite set like the final Klein quartet? Please reply ... because I looked at this topic and I came with a group like the Klein Group, but it is not final. I want you to help me publish this paper.
I do not have a link ...but i have this paper ...I wrote this paper myself ... I searched a lot on the Internet and I didn't find it like that ... I want to show it to a professor who specializes in abstract mathematics now!
Great video.
All of a sudden these separate ideas all combined together.
Thanks a lot Math sorcerer.
well, vier (German four ) sounds like feel or fear in English.
Good video. The Klein 4 group can be generated by the set of 180-degree rotations about the axes x,y, and z in three dimensional space. While the Klein 4 group is finite and commutative, the group of rotations is infinite and non-commutative. Is the group generated by the standard quaternion basis the same as the group generated by 90-degree rotations about the x, y, and z axes?
is improper subgroup.....
of the klein 4? no it's not, remember H is a proper subgroup of G if H is a subgroup and H != G. So is a subgroup of the klein 4, but it's not the whole group ,so it's proper
@@TheMathSorcerer as far as I know If H is a subgroup of G and H=G or H= singleton set of identity element, then it's improper subgroup ...
And here (e) is Identity ,so it's improper subgroup.
Plz correct me if i am wrong..
Some authors of books use that as the definition. Depends on the books . I think it's more common to exclude the set with just the identity as this way the definition agrees with the set definition of a proper subset.
Surprised no one's mentioned the set {1,3,5,7} and the binary operation multiplication modulo 8 is the Klein 4 group. Lots of isomorphisms on this one
The German word "vier" (meaning 4) rhymes with the English "deer". Just put an "f" infront.
Very similar words are also:
- fear (tone down the a)
- fierce (stop before the z loud)
Awesome!!
When you write out a group's Caley table, you'll start to see that it resembles a game of Sudoku
|e a b c
--------------
e| e a b c
a| a e c b
b| b c e a
c| c b a e
See how in each row and each column there are no duplicate elements
and every row has to have an e
So it's interesting to work out all possible groups in an abstract sense
You could use 0,1,2,3 instead of e,a,b,and c of course. Also the left most column and top most row always have to be the same for the identity element property to hold, so in this case there are 9 squares to play with
+The Math Sorcerer I really don't know why there are only 17 likes, this video is amazing, and all the work you did on Abstract Algebra is enormous and extremely valuable. Thank you very much! I have an exam tomorrow and I am revising using your videos. Thank you again!
+Suleyman Orazgulyyev I am glad it helped!!
more now!!
what about its Cailey graph?this video motivated me to think ...
Is there an infinite set like the final Klein quartet? Please reply ... because I looked at this topic and I came with a group like the Klein Group, but it is not final. I want you to help me publish this paper.
I'm not familiar with this sorry. Do you have a link?
I do not have a link ...but i have this paper ...I wrote this paper myself ... I searched a lot on the Internet and I didn't find it like that ... I want to show it to a professor who specializes in abstract mathematics now!
Why isnt the inverse of a in the cyclic subgroup generated by a?
This is 4 months later, but:
I believe a is its own inverse, since a*a = 1
=e, sorry.
@@swng314 Its 3 years late, but it helped me, so thanks
V for Viergruppen (four group)
:)
I want to thank you SINCERELY! Cannot thank you enough! /bow
np at all,glad it helped someone out there!!
In German, it's called the "kleinische Vierergruppe" -- pronounced: KLEIN-ish-uh FEAR-er-group-uh (stressing the capitalized syllables).
Great video!
thanks!!!!!!!
Thanks sir
I found the answer that i need
The diagram representation.
awesome so happy it helped!
Thank you
Nice video, Thank you.
BTW: Vier is pronounced like the English word "fear"
cool good to know!!!
vieregrouppe
indeed!
Very helpful! Thanks a lot!
Thank you so much
Thanks!
😄😄
Sir g group theory ki book jonhn - b - ka lecture tu upload karo hamari help ho jaya gi
6:25...let me help you. Pronounce it like fear and add an "er" . That's how to pronounce Vierer.
thx man:)
vier = 4 auf Deutsche.. pronounced feAR...
Thanks!
😄