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A unique factorisation domain is an integral domain $R$ which has unique factorisation. That is, every $x\in R$ has a unique decomposition into the form $\prod p_i$ where the $p_i$ are prime, up to order and units.

I find it quite strange that there is no analogue of this concept in group theory. Usually, concepts in ring theory use the addition that is defined in the rings somehow. But in this case, the definition of unique factorisation seems to be a purely multiplicative concept, with no reference to addition whatsoever. So why do we not define the concept of unique factorisation in, say, groups as well? A possible reason I can think of is that if we just took a UFD $R$ and threw away addition, then $(R,\times)$ might not be a group. An unfortunate example of this is that the integers are not a group with respect to multiplication, so it would not be a "unique factorisation group", even if we were to define the concept.

But this does not seem convincing enough of a reason to me to completely abandon the idea. Even if we really wanted to include $\mathbb Z$ inside our new concept of a unique factorisation "group", then why not weaken the group axioms to not require inverses? Still, we do not need to introduce a whole new operation of addition to make sense of the concept. What am I missing here? Why isn't unique factorisation defined this way? What does defining addition contribute?

YiFan Tey
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    Search unique factorization semigroup: https://www.google.com/search?client=ubuntu&channel=fs&q=unique+factorization+semigroup&ie=utf-8&oe=utf-8 – Ethan Bolker Mar 20 '19 at 13:03
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    The first and most obvious thing that comes to mind is Bezout's identity. Unfortunately this is relevant only when the ring is a PID as opposed to some other UFD. We then get the gcd of two elements as a generator of the ideal the two elements generate together. This business of describing ideals obviously requires addition also. – Jyrki Lahtonen Mar 20 '19 at 13:05
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    The notion (if you just copied the definitions) would be completely uninteresting for groups, since we are explicitly interested in non-invertible elements, which would leave us with no elements in a group to even work with. As mentioned above, one needs to consider semigroups. – Tobias Kildetoft Mar 20 '19 at 13:29
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    Unique factorization (and related concepts) in monoids has been studied at length, e.g. see the surveys on divisibility groups that I cite here. – Bill Dubuque Mar 21 '19 at 01:17
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    See also Franz Halter-Koch's survey Non-unique factorizations of algebraic integers for another entry point into the literature. – Bill Dubuque Mar 21 '19 at 01:31
  • Thank all of you for the kind responses! I will read up more on this subject. – YiFan Tey Mar 21 '19 at 01:54

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At least in the study of infinite, finitely generated groups, there is something of an analogue! Many such group theorists like to think about their groups as metric spaces, where the metric is the word metric: if you have a finite set $S$ of generators for a group $G$, the word length of an element $g$, $\|g\|_S$ is the minimum number of elements of $S$ and their inverse present in an equality $g = s_1s_2\dots s_\ell$. The word metric is defined as $$d_S(g,h)=\|gh^{-1}\|_S.$$

The analogue of a UFD in this context is a group $G$ and generating set $S$ with respect to which you have a language of geodesic normal forms, that is, for each $g$ you have a nice expression of $g$ as a product of elements in $S$ and their inverses with minimum (geodesic) word length.

An example is $\mathbb Z\oplus\mathbb Z$ with generating set $\{(1,0),(0,1)\}$. There the normal form for $g$ amounts to expressing it in coordinates as $g = (m,n)$ for some integers $m$ and $n$.

Groups with languages of geodesic normal forms (or various notions close to this) are often a little easier to get a handle on algebraically, making them nicer to study. But, I want to point out that this is a much weaker condition than that of a UFD. For example, there's no analogue of the Chinese Remainder Theorem for all groups with this property.

Rylee Lyman
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