Groups and Subgroups

Intro to Proofs · Axiom Academy

From four axioms to Lagrange's theorem — building the structure of a group and watching its subgroups slice it into equal pieces. A group packages a set with a way to combine its elements so that nothing ever escapes and everything can be undone. The smallest interesting example is the clock under addition mod 4 — its full Cayley table records every sum a+b . Closure: for all , the result a*b is again in G . Associativity: (a*b)*c = a*(b*c) for all . Identity: some has e*a = a*e = a for every a . Inverses: each a has an with a*a^ -1 = a^ -1 *a = e . 2. Cyclic Groups: One Element Builds Everything Some groups are generated by a single element. Start at the identity 0 in and keep adding the generator g = 1 . Each step lands on the next clock position, and after six steps you have visited every element and returned home. every element is a power (here, a multiple) of one generator 3. Subgroups: A Group Hiding Inside A subgroup is a subset that is itself a group under the same operation. Inside , the set is closed: 3+3 = 0 stays inside, 0 is the identity, and each element is its own inverse. Watch H light up and stay self-contained. , so H is non-empty and anchored at e . — every sum of members lands back in H . and -0 = 0 , so undoing never leaves H . It already holds in G , so we never re-check it. 4. Lagrange's Theorem: Equal Slices

This is the written version of the interactive lesson above. See the full Intro to Proofs course.