# Difference between revisions of "Talk:2028: Complex Numbers"

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Shouldn't the description of a group involve ''two'' operations? There is a binary operation that gloms two things together to make a new thing, but there's also a unary operation that takes only one thing and makes a new thing -- the inverse. Without the unary operation, you only have a {{w|semigroup}}.[[Special:Contributions/108.162.215.160|108.162.215.160]] 09:40, 5 August 2018 (UTC) | Shouldn't the description of a group involve ''two'' operations? There is a binary operation that gloms two things together to make a new thing, but there's also a unary operation that takes only one thing and makes a new thing -- the inverse. Without the unary operation, you only have a {{w|semigroup}}.[[Special:Contributions/108.162.215.160|108.162.215.160]] 09:40, 5 August 2018 (UTC) | ||

+ | : No. The inverse operation arises as a consequence of the fact that it's a group. A group satisfies four conditions: 1. it is closed under the operation, 2. the operation is associative 3. there is an identity e such that a op e = e op a = a. 4. For every element a, there is a unique element b such that a op b = b op a = e. The inverse function falls out as a result of conditions 3 and 4 [[User:Jeremyp|Jeremyp]] ([[User talk:Jeremyp|talk]]) 10:26, 6 August 2018 (UTC) |

## Revision as of 10:26, 6 August 2018

I assume this is strictly a coincidence, but in reference to the title-text, I'll just mention that Caucher Birkar [the mathematician whose Fields Medal was stolen minutes after he received it in Rio de Janeiro on Weds (1Aug2018)] received the award for work in algebraic geometry. Arcanechili (talk) 16:34, 3 August 2018 (UTC)

- Perhaps it's causal not coincidental. Medal theives and perhaps Randall might read the news also. [[1]] 162.158.79.209 00:34, 4 August 2018 (UTC)

I've added a basic description of Abelian groups in the title text, and that's about as much as I know about such topics. I'm not sure what a "meta-Abelian group" is, is that an Abelian group of other groups? Also, could someone add basic descriptions of algebreic geometry and geometrical algebra? 172.68.94.40 18:42, 3 August 2018 (UTC)

In the title text, since groups are a concept within mathematics, it seems odd to consider mathematics as a whole forming any sort of group within itself, which I suspect is the first part of the pun. Secondly, since groups involve the commutative property, I think the last part is a pun about the order of the words algebra and geometry, as if they're commutative themselves! Ianrbibtitlht (talk) 19:19, 3 August 2018 (UTC)

- I meant to say 'abelian' groups involve the commutative property, and the meta prefix is referring to the fact that it's about the names rather than the mathematical details - i.e. commutative in metadata only. Ianrbibtitlht (talk) 19:24, 3 August 2018 (UTC)

- I guess the joke is that informally mathematicians form a
*group*(a number of people classed together), what would strictly be a*set*in mathematics. While in mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies specific conditions. --JakubNarebski (talk) 21:18, 3 August 2018 (UTC)

It's a false dilemma. Complex numbers *are* vectors ( is a two-dimensional -vector space, and more generally every field is a vector space over any subfield), but that doesn't change anything about the fact that is by definition a square root of -1. Zmatt (talk) 20:38, 3 August 2018 (UTC)

Fun factoid: not only is the unique proper field extension of finite degree over (since is algebraically closed), but the converse is true as well: is the only proper subfield of finite index in . They're like a weird married couple. Zmatt (talk) 20:53, 3 August 2018 (UTC)

Altho there are no "meta-abelian" groups there are metabelian groups. If xy=yx then the commutator [x,y]=xyx^{-1}y^{-1}=1. The group generated by the commutators -- the commutator subgroup -- is thus a measure of how far a group is from being abelian. A metabelian group is a nonabelian group whose commutator subgroup is abelian. Thus a metabelian group is one made of a stack of two abelian groups. It is "meta-abelian" in that sense. A standard example is the group of invertible upper-trianglular matrices. The commutators all have 1s on the diagonals.

- One should note that the concept of complex numbers actually is older than vector spaces. So while it is true that complex numbers are a cool variant of vectors, historically that's not true, because vectors were more or less unknown when complex numbers were used for the first time. --162.158.90.6 09:59, 4 August 2018 (UTC)

Shouldn't the description of a group involve *two* operations? There is a binary operation that gloms two things together to make a new thing, but there's also a unary operation that takes only one thing and makes a new thing -- the inverse. Without the unary operation, you only have a semigroup.108.162.215.160 09:40, 5 August 2018 (UTC)

- No. The inverse operation arises as a consequence of the fact that it's a group. A group satisfies four conditions: 1. it is closed under the operation, 2. the operation is associative 3. there is an identity e such that a op e = e op a = a. 4. For every element a, there is a unique element b such that a op b = b op a = e. The inverse function falls out as a result of conditions 3 and 4 Jeremyp (talk) 10:26, 6 August 2018 (UTC)