Algebraic Groups (GroupFinder)

# Algebraic Groups (Group Finder) – IIG Technology

Android Java application that checks if a given finite structure is a group
and explores its properties.

## Structures
Zₙ, Zₙ*, {1,…,n−1} with multiplication mod n, Dₙ, Sₙ (n ≤ 5), Aₙ (n ≤ 5), Q₈,
Zₘ × Zₙ and its own Cayley table (up to 64 elements).

## Analysis
- Axioms: closedness, associativity, neutral element, converses – with counterexample in case of failure
- Isomorphic type: complete classification of abelian groups (elementary divisors and
invariant factors); recognition of S₃, D₄, Q₈, A₄, D₆, Dic₃, S₄, A₅,
groups of order pq
- Abelianity, cyclicity and generators, exponent, center, conjugacy classes,
commutant, solvability, simplicity
- All subgroups (for order ≤ 64) with marking of normal ones (⊴)
- Cayley color table with selected subgroup coloring
- SQLite history, languages: BG / EN / FR / DE / ES

## Custom table format
```
e a b c ← elements
e a b c ← e∘e, e∘a, e∘b, e∘c
a e c b
b c e a
c b a e
Here's how to read the result, section by section, with the example D₄: the symmetries of a square, 8 elements. The rotations are e, r, r², r³, and the reflections are s, sr, sr², sr³. The application outputs:
D₄ |G| = 8

AXIOMS ​​OF THE GROUP
✔ Closedness
✔ Associativity
✔ Neutral element: e
✔ Every element has an inverse

RESULT: this is a group of row 8

PROPERTIES
• Isomorphic to: D₄
• Abelian: not (r∘s ≠ s∘r)
• Cyclic: no
• Exponent: 4
• Center Z(G): {e, r²} (row 2)
• Cohesion classes: 5 (sizes: 1, 2, 1, 2, 2)
• Kommutant [G,G]: ed 2
• Resolvable: yes
• Simple group: no

ROWS OF ELEMENTS
word = 1 → 1
words = 2 → 5
words = 4 → 2
e: 1, r: 4, r²: 2, r³: 4, s: 2, sr: 2, sr²: 2, sr³: 2

SUBGROUPS
Subgroups: 10, of which normal: 6
In order: |H|=1 → 1, |H|=2 → 5, |H|=4 → 3, |H|=8 → 1
1. Group axioms
The application checks the four axioms sequentially, directly against the Cayley table:
Closedness. Every product a∘b must be an element of the set.
Associativity. It is verified that (a∘b)∘c = a∘(b∘c) for all n³ triples.
Neutral element. We are looking for e for which e∘x = x∘e = x for every x.
Inverse elements. For each x, find y with x∘y = y∘x = e.
In violation, you get a specific counterexample, not just a "no." For example, for {1,…,7} with multiplication mod 8:
✘ Closed: 2 ∘ 4 is not in the set
The reason is that 2·4 = 8 ≡ 0 (mod 8), and 0 is not in the set. If the closure is not satisfied, the remaining axioms are not verified because the operation is not well defined at all.
2. Properties (only if it is a group)
Isomorphic to. This is the "true identity" of the group: two groups with the same tableau after renaming the elements are the same group. For example, {1,…,6} with multiplication mod 7 is recognized as Z₆, even though its elements are numbers, not remainders under addition.
Abelian. Checks whether a∘b = b∘a always. If not, shows a pair that does not commute: here r∘s = sr³, and s∘r = sr.
Elementary divisors and invariant factors (only for abelian groups). Every finite abelian group is a product of cyclic groups, and these are two standard notations of this decomposition. For example, for Z₄ × Z₆:
Elementary divisors: Z₂ × Z₄ × Z₃. This is a decomposition of cyclic groups of prime order.
Invariant factors: Z₂ × Z₁₂. Here each factor divides the next (2 | 12).
Both entries describe the same group. They are calculated from the number of elements with a rank dividing pᵏ.
Cyclic and generating. A group is cyclic if one element generates all the others. For example, Z₁₂ has generators 1, 5, 7, 11, that is, the numbers coprime to 12.
Exponent. The smallest k for which xᵏ = e for every x. It is the LOC of the rows of the elements, here LOC(1, 2, 4) = 4. If the exponent is equal to the row of the group, the group is cyclic.