#### Neutrons, Protons and Electrons - Isotopes

An atom is made up of a nucleus, itself made up of protons and neutrons held together by the strong nuclear force, and electrons, which form a 'cloud' and the nucleus. The following points must be made: The number of electrons is equal to the number of...

https://astarmathsandphysics.com/ib-physics-notes/atomic-and-nuclear-physics/1222-neutrons-protons-and-electrons-isotopes.html#### 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#### Field Axioms

A field is a set that is a commutative group with respect to two compatible operations, addition and multiplication, with "compatible" being formalized by distributivity, and the caveat that the additive identity (0) has no multiplicative...

https://astarmathsandphysics.com/university-maths-notes/abstract-algebra-and-group-theory/1693-field-axioms.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#### Fission Reactors

The basic elements of a fission reactor are: reactor core for holding fission material or fuel moderator for slowing fast neutrons control rods holding neutron absorbers to control rate of fission monitoring system containing devices and indicators of...

https://astarmathsandphysics.com/o-level-physics-notes/164-fission-reactors.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#### Joule's Experiment

James Joule demonstrated the link between mechanical work done and temperature change. He conducted a series of experiments showing that doing work resulted in an increase in temperature proportional to the amount of work done, allowing for...

https://astarmathsandphysics.com/ib-physics-notes/the-history-and-development-of-physics/1314-joule-s-experiment.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#### The Cyclic Notation For Permutation Groups

Consider the set of elements {1,2,3,,...,n}. We may reorder these, choosing the first in n ways, the second in ways and so on. There are n! Possible rearrangements of the set {1,2,3,,...,n}. The set of all possible rearrangements is call the...

https://astarmathsandphysics.com/university-maths-notes/abstract-algebra-and-group-theory/1716-the-cyclic-notation-for-permutation-groups.html

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