Talk:One-parameter group

What about the definition: the set of algebraic group homomorphisms from $\mathbb C^*$ to $G$? 129.215.104.100 (talk) 11:55, 19 September 2012 (UTC)

Usage

The phrase One-parameter group is often used to mean one-dimensional Lie group. At present, this article notes that a particular group homomorphism is being designated by the phrase, so that this particular kind of group is not a group. The structure of a "one-dimensional Lie group" is no different than that of the real line as a group under addition, so its features don't inspire an article. The analysis of the concept presently presented would be confusing to a general reader and verges on meta-mathematics. Given that the topic has a significant literature, there may be sources to fill the vacuum and counter the obfuscation of the non-group group.Rgdboer (talk) 01:04, 10 January 2015 (UTC)

A link to usage by Sophus Lie in 1893 has been posted.Rgdboer (talk) 03:11, 10 January 2015 (UTC)

That's definitely a right step. Nice work! -- Taku (talk) 04:01, 10 January 2015 (UTC)

The definition (as given in the lead) agrees with all literature I have come across. It is also usually pointed out that it is, in fact, not a group. The topic of this homomorphism does motivate an article imo, while the group (ℝ, +), of course, does not. YohanN7 (talk) 17:31, 10 January 2015 (UTC)

Not a group

"Discussion"

That means that it is not in fact a group,<ref [http://math.stackexchange.com/questions/353781/one-parameter-group-not-a-group-why "One-parameter group not a group? Why?"], ''[[Stack Exchange]]'' Retrieved on 9 January 2015. /ref> strictly speaking;

That is, we start knowing only that

$\varphi (s+t)=\varphi (s)\varphi (t)$

where $s$, $t$ are the 'parameters' of group elements in $G$. We may have

$\varphi (s)=e$, the identity element in $G$,

for some $s\neq 0$. This happens for example if $G$ is the unit circle and

$\varphi (s)=e_{}^{is}$.

In that case the kernel of $\varphi$ consists of the integer multiples of $2\pi$.

Therefore a one-parameter group or one-parameter subgroup has to be distinguished from a group or subgroup itself, for the three reasons

1. it has a definite parametrization,
2. the group homomorphism may not be injective

_______

• Much of the "Discussion" is moved here since its only reference is to Stack Exchange, not a WP:Reliable source. Ambiguous use of e is bothersome. Two snippets have been preserved.

— Rgdboer (talk) 00:47, 7 March 2019 (UTC)