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Abstract Algebra: Abelian Groups of Order 108 and 32, Ring Examples, Ring Definition
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Up to isomorphism, there are 6 Abelian groups of order 108=2^2*3^3 and 7 Abelian groups of order 32=2^5. We also work through ring examples and construct the definition of a ring based on the examples.
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⏱️TIMESTAMPS⏱️
(0:00) Abelian groups of order 108=2^2*3^3 (up to isomorphism)
(4:30) The exponent 2 can be partitioned in 2 ways.
(5:24) The exponent 3 can be partitioned in 3 ways.
(5:57) There are 5 partitions of 4 and 7 partitions of 5.
(7:47) Abelian groups of order 32=2^5 (up to isomorphism).
(10:50) There are 7*7=49 Abelian groups of order 2^5*3^5=7776.
(12:15) Ring theory
(13:26) Example 1: Z is a commutative ring with unity 1 (multiplicative inverses), under ordinary addition and multiplication. The only units with multiplicative inverses are 1 and -1.
(21:51) Example 2: 2Z (the even integers) is a commutative ring without unity.
(25:43) Example 3: M2(R) the ring of 2x2 matrices with real number entries under ordinary matrix addition and matrix multiplication.
(34:52) Example 4: Z6 under addition and multiplication mod 6.
(42:08) Zero divisors can exist in Rings (where the product of two nonzero elements is zero)
(43:48) Definition of a ring.
(52:34) The multiplicative identity is denoted by 1 (though this could be the identity matrix or the constant polynomial p(x)=1)
(55:31) Unity of a ring is the multiplicative identity, when it exists
(56:04) Units of a ring with unity are those with multiplicative inverses.
(57:35) 0 is never a unit (the additive identity never has a multiplicative inverse)
(58:46) A field is a commutative ring with unity whose nonzero elements form a group under multiplication (all the nonzero elements are units).
AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.
Links and resources
===============================
⏱️TIMESTAMPS⏱️
(0:00) Abelian groups of order 108=2^2*3^3 (up to isomorphism)
(4:30) The exponent 2 can be partitioned in 2 ways.
(5:24) The exponent 3 can be partitioned in 3 ways.
(5:57) There are 5 partitions of 4 and 7 partitions of 5.
(7:47) Abelian groups of order 32=2^5 (up to isomorphism).
(10:50) There are 7*7=49 Abelian groups of order 2^5*3^5=7776.
(12:15) Ring theory
(13:26) Example 1: Z is a commutative ring with unity 1 (multiplicative inverses), under ordinary addition and multiplication. The only units with multiplicative inverses are 1 and -1.
(21:51) Example 2: 2Z (the even integers) is a commutative ring without unity.
(25:43) Example 3: M2(R) the ring of 2x2 matrices with real number entries under ordinary matrix addition and matrix multiplication.
(34:52) Example 4: Z6 under addition and multiplication mod 6.
(42:08) Zero divisors can exist in Rings (where the product of two nonzero elements is zero)
(43:48) Definition of a ring.
(52:34) The multiplicative identity is denoted by 1 (though this could be the identity matrix or the constant polynomial p(x)=1)
(55:31) Unity of a ring is the multiplicative identity, when it exists
(56:04) Units of a ring with unity are those with multiplicative inverses.
(57:35) 0 is never a unit (the additive identity never has a multiplicative inverse)
(58:46) A field is a commutative ring with unity whose nonzero elements form a group under multiplication (all the nonzero elements are units).
AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.
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