• Sylow's Second Theorem

    Sylow's Second Theorem Let be a finite group of order n and let be a prime dividing then the number of distinct Sylow – subgroups (remember that if is the highest power of dividing then the Sylow – subgroup of is that subgroup which has order ). of is...

    https://astarmathsandphysics.com/university-maths-notes/abstract-algebra-and-group-theory/1712-sylow-s-second-theorem.html

  • Properties of Matrix Multiplication

    When two matrices and are multiplied to produce a third matrix the entry in the ith row and jth column, labelled can be considered as a dot product. If the matrix is considered to be made up of row vectors and the matrix is considered to be made up of...

    https://astarmathsandphysics.com/ib-maths-notes/matrices/1006-properties-of-matrix-multiplication.html

  • Properties of Permutations

    Every permutation can be written as a cycle or a product of disjoint cycles. This follows by considering the effect of a sequence of permutations on each member of the set Each element i will end up after a sequence of permutations in some other...

    https://astarmathsandphysics.com/university-maths-notes/abstract-algebra-and-group-theory/1705-properties-of-permutations.html

  • Group Actions

    Let be a set and let be a group who elements act on the set A left group action is a function such that: for all where is the identity element in for all and Right actions are similarly defined. From these two axioms, it follows that for every the...

    https://astarmathsandphysics.com/university-maths-notes/abstract-algebra-and-group-theory/1696-group-actions.html

  • Second Principle of Induction Derived From the Well Ordering Principle

    Second Principle of Mathematical Induction Let \[P(n)\] be a proposition depending on an integer \[n\] . If 1) \[P(n_0)\] is true for some integer \[n_0\] 2) For \[k \gt n_0\] \[P(n_0), \; P(n_0+1), \; P(n_0+2),..., \; P(k)\] are true then \[P(n)\] is...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5214-second-principle-of-induction-derived-from-the-well-ordering-principle.html

  • Integral Domains

    An integral domain is a commutative ring with no zero-divisors: or Examples The ring is an integral domain. (This explains the name.) The polynomial rings and are integral domains. (Look at the degree of a polynomial to see how to prove this.) The ring...

    https://astarmathsandphysics.com/university-maths-notes/abstract-algebra-and-group-theory/1697-integral-domains.html

  • Permutationi Groups

    A permutation group is a group of order whose elements are permutations of the integers The set of all permutations is labelled and called the symmetric group. A permutation group labelled is usually a subgroup of the symmetric group. As a subgroup of...

    https://astarmathsandphysics.com/university-maths-notes/abstract-algebra-and-group-theory/1702-permutationi-groups.html

  • Incompressible Fluids

    If a fluid is incompressible and is the velocity vector field that describes the velocity of the fluid at each point then This equation follows from the continuity equation First write the continuity equation as Since the fluid is incompressinble There...

    https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1570-incompressible-fluids.html

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