Group can be determined by how elements multiply with each other

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Discussion Overview

The discussion centers around the idea that all properties of a group can be determined by how its elements interact through multiplication. Participants explore the implications of this notion, particularly in relation to the structure of finite groups and the challenges of analyzing multiplication tables.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that all properties of a group can be derived from its multiplication table.
  • One participant outlines the axioms of group theory, suggesting that they can all be satisfied for a finite group as indicated by its multiplication table.
  • Another participant notes the difficulty of determining whether a group of order 60 is simple by examining its multiplication table, drawing an analogy to calculating volumes from equations.
  • Concerns are raised about the practical challenges of verifying associativity from a multiplication table, especially for groups with a larger number of elements.
  • One participant reiterates that a group is fundamentally defined by its elements and operation, acknowledging the theoretical possibility of constructing a multiplication table, even for infinite groups.

Areas of Agreement / Disagreement

Participants generally agree that the multiplication table contains significant information about a group's properties, but there are differing views on the practicality and feasibility of deriving all properties from it, especially for larger groups.

Contextual Notes

Participants highlight limitations related to the complexity of verifying group axioms from multiplication tables, particularly for groups of larger orders, and the challenges posed by infinite groups.

tgt
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Given a group, it seems everything (all properties) about the group can be determined by how elements multiply with each other. Correct? For example, everything about the group could be read off from the multiplication table.
 
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1. Closure. For all a, b in G, the result of the operation ab is also in G.
2. Associativity. For all a, b and c in G, the equation (ab)c = a(bc) holds.
3. Identity element. There exists an element 1 in G, such that for all elements a in G, the equation 1a = a1 = a holds.
4. Inverse element. For each a in G, there exists an element b in G such that ab = ba = 1, where 1 is the identity element.

After looking at the axioms one at a time, it sure looks like they all can be satisfied for a finite group in a multiplication table.
 
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well it is hard to look at the multiplication table of a group of order 60 and decide whether it is simple or not, but in principle yes, everything is determined by the multiplication table.

but that's like saying everything about a solid is determined by its equation, but you still may have trouble computing its volume from that knowledge. e.g. the length of the graph of y = x^3 is determined by that equation, but what is that length from x=0 to x=1?
 


but still, even if you don't have a group of a that big order, say you have a group of 6 elements, then it would be a pain to check the associativity from the operation table. since there would be x nr of cases, where x is the nr of permutations with repitition.
 


tgt said:
Given a group, it seems everything (all properties) about the group can be determined by how elements multiply with each other. Correct? For example, everything about the group could be read off from the multiplication table.
Well, yeah! A group is defined by its members and its operation- there is nothing else. Of course that operation is not always defined in terms of a multiplication table, but you could, theoretically, write one. (That might be very difficult in the case of infinite groups!)
 

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