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I'm trying to form a ring (or ring-like structure) out of the set of all finite groups.

Has anyone created/encountered an operation "+" before with the follow properties.

Let $G_1, G_2$ be finite groups and |G| denote the size of G, then

$$ |G_1| + |G_2| = |G_1 + G_2|$$

Where the left hand side is addition amongst natural numbers and the right hand side is our "abstract group addition"

And let $\times$ denote the direct product. Then:

$$ G_1 \times ( G_2 + G_3) = (G_1 \times G_2) + (G_1 \times G_3) $$

The "+" ideally would be commutative and associative, but I'm not sure if such an operator can necessarily exist.

My goal is to try to denote a notion of "rational groups" and dedekind cuts to groups to see if I can create a "fractional group theory" so to speak.

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  • $\begingroup$ That would fail the distributive multiplicative identity. Since the RHS would be cyclic, but the LHS would a direct product of some group and a cyclic group $\endgroup$
    – Sidharth Ghoshal
    Jun 21, 2017 at 5:19
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    $\begingroup$ BTW, great story about linear optimization and drug dealers! $\endgroup$
    – Asvin
    Jun 21, 2017 at 5:45
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    $\begingroup$ Technicality: "All finite groups" is a class, not a set. "All isomorphism classes of finite groups" is a set. $\endgroup$
    – zibadawa timmy
    Jun 21, 2017 at 6:00
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    $\begingroup$ Every homomorphism from this additive magma to an abelian group would factor through $\mathbf{Z}$. Indeed, denoting by $c_n$ the image of the cyclic group of order $n$, we would have $pc_1=c_p$ for all prime $p$. Then I claim that for every $n\ge 1$, if $g$ is the image of a group of order $n$, then $g=nc_1$. Proof by induction on $n$; it's clear for $n=1$. For given $n\ge 2$, there is a prime $p$ with $n\le p<2n$; the case $n$ prime being ok, we can suppose $p>n$. Then $g+c_{p-n}=c_p$, so $g+(p-n)c_1=pc_1$, thus $g=nc_1$. So the group associated with such a structure would be quite poor. $\endgroup$
    – YCor
    Jun 21, 2017 at 8:11
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    $\begingroup$ A related notion is en.wikipedia.org/wiki/Distributive_category. Note it mentions Groups do not form such a category, but without explanation. It looks like your structure would be akin to a distributive category structure on Groups that is compatible with the distributive structure on (finite) Sets via the forgetful functor. $\endgroup$
    – PrimeRibeyeDeal
    Jun 21, 2017 at 14:22

1 Answer 1

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I believe that we can get a commutative, associative operation as follows. Let $G=A\times B$ and $H=A\times C$, where $B$ and $C$ have no common factor (when written as a direct product of indecomposable groups of order greater than one). Define $G+H=A\times D$, where $D$ is a product of cyclic groups of prime order such that $|D|=|B|+|C|$.

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