Re: It it a good idea to use the term 2-group outside of its use in group thoery?

Re: It it a good idea to use the term 2-group outside of its use in group thoery?

[Toby Bartels]

Ronnie Brown wrote:

>I would  like to raise an objection  to using the term `2-group' as on nlab and elsehere since for the group theorists this has a specialised meaning: See the following wiki entry, especially the first 2 words:

>"In mathematics, given a prime number p, a p-group is [...]"

That the first 2 words are "In mathematics" rather than
"In group theory, a branch of mathematics," means nothing.
It's not like the Wikipedians had a discussion about it
and determined that p-groups appear throughout mathematics.

You do raise a good point, though.  The term '2-group' is a special case
of both 'p-group' and 'n-group', and these mean very different things.

I wouldn't want to give up 'n-group', so I find '2-group' appropriate
when (as on the n-Category Lab) one is discussing n-groups as well.
But in your example about the structure of finite crossed modules,
one can simply say 'crossed module', making a note that some literature
calls a crossed module a '2-group' (or even 'strict 2-group').

>"[...] Such groups are also called primary."
>there are claims that crossed modules, for example, can be thought of as `2-dimensional groups'

In extreme cases, these show the way: both 'p-group' and 'n-group'
are abbreviations, for 'p-primary group' and 'n-dimensional higher group'.
So one can always use the full name or specify which usage one's paper follows.


--Toby

It it a good idea to use the term 2-group outside of its use in group thoery?

[Ronnie Brown]

I would  like to raise an objection  to using the term `2-group' as on nlab and elsehere since for the group theorists this has a specialised meaning: See the following wiki entry, especially the first 2 words: 

"In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element. Such groups are also called primary." 

I feel we should try to  avoid and even  to reduce  confusion, especially as there are claims that crossed modules, for example, can be thought of as `2-dimensional groups' (I agree with this, of course!); there are nice crossed modules M \to P in which M and P are 2-groups in the group theoretic sense! 

My favourite example is 

\mu: Z_2 \times Z_2 \to Z_4 

in which Z_4 acts by the twist (of order 2), and \mu maps each factor Z_2 injectively into Z_4. This crossed module has non trivial k-invariant. I think Johannes Huebschmann first observed this. 

So an example oriented approach to crossed modules could well need the term p-group in its standard group theoretic usage. Some examples of finite crossed modules are in 

R. Brown and C.D. Wensley, `Computation and homotopical applications
of induced crossed modules', J. Symbolic Computation 35 (2003)
59-72.

However I think one can be happy with the well established term 2-groupoid. 

I would just like this point to be discussed: terminology is important, and confusing an established  use might raise hackles unnecessarily. 

Ronnie