We present the quaternion group. This is an important example of a non-abelian group of small order. www.michael-penn.net www.randolphcollege.edu/mathem...
@@feridetaskn3121 All subgroups of quaternion group are normal. It is easy to see this: The whole group is normal trivially, any subgroup of order 4 would be normal because it is of index 2 and finally check that all subgroup of order 2 are normal. (Just draw the cayley table!) Quaternions are not abelian because ij=k but ji=-k.
Could I get a clarification on something? For i, j, and k are we supposed to treat them like a complex number or basis vector? The cyclic subgroups look to me like powers of complex numbers but I've normally heard of quaternions described as a 4d vector space with a vector part and scalar part.
I think the answer is both. For quaternions in general, i j and k are basis vectors, but they’ve been endowed with a rule to multiply them. In this video though, the focus is on the quaternion group, not quaternions in general. The elements {1, -1, i, j, k, -i, -j, -k} form a group under the quaternions’ multiplication rule. This is a more restricted view though, since we aren’t talking about any other quaternions that you could get by scaling and adding these ones.
Question about showing (-i)^2 = -1. You wrote (-i)^2 = (-i)(-i) = (-1)^2 i^2. But doesn't that last step involve commuting i and -1, which is not necessarily allowed in this group?
The only elements that don't commute are i, j and k between themselves. -1 is a scalar and thus commute with any element in H. (More formally the quarternions H is a 4D algebra over the real numbers, thus scalars aka real numbers commute with anything, while the basis "vectors" i, j and k do not commute with each other, but they do anti-commute, so you still have some semblence of structure)
great as always!! can u please post a video explaining what are primitive roots. u have made many videos related to primiitve roots.But I couldn't find one which explains what exactly primitive roots are . Thanks!!
This also disproves the converse of the following fact: Every subgroup of an abelian group is normal. It's quite a cool group!
@@feridetaskn3121 All subgroups of quaternion group are normal. It is easy to see this: The whole group is normal trivially, any subgroup of order 4 would be normal because it is of index 2 and finally check that all subgroup of order 2 are normal. (Just draw the cayley table!)
Quaternions are not abelian because ij=k but ji=-k.
I feel happy that I can understand a maths video for once
Could I get a clarification on something? For i, j, and k are we supposed to treat them like a complex number or basis vector? The cyclic subgroups look to me like powers of complex numbers but I've normally heard of quaternions described as a 4d vector space with a vector part and scalar part.
I think the answer is both. For quaternions in general, i j and k are basis vectors, but they’ve been endowed with a rule to multiply them. In this video though, the focus is on the quaternion group, not quaternions in general. The elements {1, -1, i, j, k, -i, -j, -k} form a group under the quaternions’ multiplication rule. This is a more restricted view though, since we aren’t talking about any other quaternions that you could get by scaling and adding these ones.
How the arithmetic operations addition, subtraction, multiplication and division are performed within that group?
Fascinating. Thank you.
Question about showing (-i)^2 = -1. You wrote (-i)^2 = (-i)(-i) = (-1)^2 i^2. But doesn't that last step involve commuting i and -1, which is not necessarily allowed in this group?
The only elements that don't commute are i, j and k between themselves. -1 is a scalar and thus commute with any element in H.
(More formally the quarternions H is a 4D algebra over the real numbers, thus scalars aka real numbers commute with anything, while the basis "vectors" i, j and k do not commute with each other, but they do anti-commute, so you still have some semblence of structure)
great as always!! can u please post a video explaining what are primitive roots. u have made many videos related to primiitve roots.But I couldn't find one which explains what exactly primitive roots are . Thanks!!
That is a great idea! I'll make this one of my next videos.
@@MichaelPennMath link?
Hello, fellow nerds.
Clockwise is dual to anti-clockwise.
Here's another mnemonic
I=jk (i am joking)
Huh …? Who invented this gibberish ?
🧐🧐🧐🤔🤔🤔🤨🤨🤨😡😡😡
It's a necessary part of how your phone keeps track of its 3-D rotation in space FYI.