# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Finite group: auditing the group axioms from a complete Cayley table](/en/derivations/derivation-mko-finite-group-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-finite-group)

Two complete operation tables are given. (A) For G={0,1,2,3} with table T, prove that (G,*) is a finite group: closure, associativity for all 4³=64 triples, a two-sided identity and two-sided inverses. (B) K={0,1,2} with table ∘ is symmetric, has a two-sided identity and two-sided inverses for every element, yet violates associativity at the triple (1,1,2), so it is not a group; also show that all 19 triples containing 0 pass, so sampling would misjudge it.

Finite group

derivation-mko-finite-group-01



Concept guide



## [A complete six-element action on four points](/en/derivations/derivation-mko-finite-group-action-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-finite-group)

The explicit 24-entry table is a valid group action, but it is neither free nor transitive.

Finite group action

derivation-mko-finite-group-action-01



Concept guide



## [Complete orbits and their partition](/en/derivations/derivation-mko-finite-group-orbit-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-finite-group)

The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation.

Orbit of a finite group action

derivation-mko-finite-group-orbit-01



Concept guide



## [All group elements fixing point zero](/en/derivations/derivation-mko-finite-group-stabilizer-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-finite-group)

Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions.

Stabilizer of a finite group action

derivation-mko-finite-group-stabilizer-01



Concept guide



## [Kernel, fibers and injectivity from C4 to C2](/en/derivations/derivation-mko-finite-group-homomorphism-kernel-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-finite-group)

For f:C4 to C2, f(a)=a mod 2, the normal kernel is {0,2}; fibers are its cosets and f is not injective.

Kernel of a finite group homomorphism

derivation-mko-finite-group-homomorphism-kernel-01



Concept guide



## [Full conjugation action and three classes](/en/derivations/derivation-mko-finite-group-conjugacy-class-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-finite-group)

The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate.

Conjugacy class in a finite group

derivation-mko-finite-group-conjugacy-class-01



Concept guide



## [Certifying the center from all commutation rows](/en/derivations/derivation-mko-finite-group-center-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-finite-group)

Z(G)={e}, directly a normal subgroup; G is nonabelian and r is not central despite commuting with itself.

Center of a finite group

derivation-mko-finite-group-center-01





1 7 results · Page 1 of 1