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) [reply ]

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) [reply ]

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

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

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) [reply ]

"Discussion"

That means that it is not in fact a group,<ref "One-parameter group not a group? Why?", Stack Exchange Retrieved on 9 January 2015. /ref> strictly speaking;

That is, we start knowing only that

${\displaystyle \varphi (s+t)=\varphi (s)\varphi (t)}${\displaystyle \varphi (s+t)=\varphi (s)\varphi (t)}

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

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

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

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

In that case the kernel of ${\displaystyle \varphi }${\displaystyle \varphi } consists of the integer multiples of ${\displaystyle 2\pi }${\displaystyle 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) [reply ]

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