Group

Dependencies: None

A group $G$ is a set $S$ along with a binary operator $*$ which satisfies certain properties:

The order of $G$, denoted as $|G|$, is the number of elements in $G$.

For $g \in G$ and $k \in \mathbb{Z}$, \[ g^k = \begin{cases} g*g*\ldots*g \; (k \textrm{ times}) & \textrm{if } k > 0 \\ e & \textrm{if } k = 0 \\ g^{-1}*g^{-1}*\ldots*g^{-1} \; (-k \textrm{ times}) & \textrm{if } k < 0 \end{cases} \]

$\operatorname{order}(g)$ is the smallest positive integer $k$ such that $g^k = e$. If such a $k$ does not exist, $\operatorname{order}(g) = \infty$.

$G$ is said to be commutative or abelian iff $\forall a, b \in G, a*b = b*a$.

The following are very common examples of abelian groups (which is trivial to prove):

Dependency for:

  1. Distribution of sum of random variables (incomplete)
  2. Inverse of a group element is unique Used in definition
  3. Condition for a subset to be a subgroup
  4. External direct product is a group
  5. Subgroup
  6. Conditions for a subset to be a subgroup
  7. Order of element in finite group is finite
  8. Order of element divides order of group
  9. Identity of a group is unique Used in definition
  10. Group element to the power group size equals identity
  11. Inverse of product of two elements of a group
  12. Isomorphism on Groups
  13. Permutation group
  14. Group of prime order is cyclic
  15. Homomorphism on groups
  16. Field
  17. Ring
  18. Coset
  19. Vector Space

Info:

Transitive dependencies: None