Properties of Group

  1. The identity element of s group is unique.
  2. The inverse of each element of a group is unique, i.e., in a group G with operation  * for every a \in G, there is only element {a^{ - 1}} such that{a^{ - 1}} * a = a * {a^{ - 1}} = e, e being the identity.
  3. The inverse a of{a^{ - 1}}, then the inverse of {a^{ - 1}} is a, i.e., {\left( {{a^{ - 1}}} \right)^{ - 1}} = a.
  4. The inverse of the product of two elements of a group G is the product of the inverse taken in the inverse order i.e.{\left( {a * b}       \right)^{ - 1}} = {b^{ - 1}} * {a^{ - 1}}{\text{ }}\forall a,b \in G.
  5. Cancellation laws holds in a group, i.e., if a,b,c are any elements of a group G, then a * b = a * c \Rightarrow b = c (left cancellation law), b * a = c * a       \Rightarrow b = c (right cancellation law).
  6. If  G is a group with binary operation  * and if a and b are any elements of G, then the linear equations a * x = b and y * a = b have unique solutions in G.
  7. The left inverse of an element is also its right inverse, i.e. {a^{ - 1}} * a = e = a * {a^{ - 1}}.

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