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The question: is it normal/expected that a group can have multiple incongruous internal structures based on changing the group operation?

Take the multiplicative group of numbers relatively prime with $30$ modulo $30$, which explicitly are $1,7,11,13,17,19,23,29.$ One usual subgroup structure is separating these group members by multiplicative order, which gives subsets as $\{1\},\{11,19,29\},\{7,13,17,23\}.$

A second separation is based on whether each number has a "construction" under a given divisor of the modulus. Without going into too much detail of this esoteric theory, here are the constructions and how they separate this same group:

$$\begin{align} 1&:1+30,16+15,81+10,25+6\\ 7&:27+10,25+12\\ 11&:81+20,5+6\\ 13&:3+10,25+18\\ 17&:2+15,27+20,5+12\\ 19&:4+15,9+10,25+24\\ 23&:8+15,3+20,5+18\\ 29&:9+20,5+24 \end{align}$$

The separation is then into subsets $\{1\},\{17,19,23\},\{7,11,13,29\}$ by "the number of distinct divisors of $30$ which have constructions for each congruence class," noting that $5\mid 30$ and $6\mid 30$ are not treated as distinct in this case, since $5\cdot 6=30$.

Follow up question: does this secondary structure appear to have any significance or use?

Extra detail for the "constructions:" the two summands may each be comprised only of powers of the primes factors of either the divisor of the modulus or else the modulus divided by this divisor. So for modulus $H$ and divisor $d$ it would be expected that for $Q\equiv b+c\mod H$ we have $b\mid d^H,d\mid b$ and $c\mid\left(\frac Hd\right)^H,\frac Hd\mid c$.

abiessu
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  • Of course, except for the case when the group is trivial or has prime order. – Moishe Kohan Dec 01 '23 at 20:54
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    To directly address your question: for any group $G$ with operation denoted $g_1 \cdot g_2$, and for any other group $H$ having the same cardinality as $G$ and with group operation denoted $h_1 * h_2$, and for any bijection $f : G \to H$, you can define another group operation on $G$ having utterly nothing to do with the operation $g_1 \cdot g_2$ by using the transport of structure formula: $$g_1 \star g_2 = f(g_1) * f(g_2)$$ – Lee Mosher Dec 01 '23 at 20:55
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    When you have a group, the term "structure" usually refers to the group operation, as the comment by LeeMosher deals with. But the examples you give are two different equivalence relations, and yes, one can easily have more than one of those on a group. – JonathanZ Dec 01 '23 at 21:12
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    For example we can transport the algebraic structure (operations) from any countable group or ring to any other countable set, e.g. $\Bbb Z$ or $\Bbb N,,$ e.g. see here and here. $\ \ $ – Bill Dubuque Dec 01 '23 at 21:32
  • @LeeMosher: I think you meant to write $$g_1 \star g_2 = f^{-1}\bigl(f(g_1) * f(g_2)\bigr).$$ – Joe Dec 01 '23 at 22:28
  • Taking the comments into consideration, I see that I have some terminology to learn and also it looks like "this is expected" is the answer to the question I mean to ask even if the terminology is wanting. Thank you commenters for the help. – abiessu Dec 02 '23 at 01:54
  • Oops, thanks @Joe. – Lee Mosher Dec 02 '23 at 14:23

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What happened here was a conflation on my part where I thought the multiplicative orders of these congruence classes modulo $30$ should correspond to the counts of distinct construction divisors associated with each congruence class. The former has roots in the subgroup structure under multiplication, but is not the subgroup structure, while the latter is the same as the subgroup structure.

abiessu
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