Blogs about: Normal Subgroup
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A finite group whose order is the product of three distinct primes has a normal Sylow subgroup of largest order
Let be a group of order where and , , and are primes. Prove that a Sylow -subgroup of is normal. Recall that some Sylow subgroup of is normal. Let , , and denote Sylow -, -, and -subgroups of , respec… more →
Project Crazy Project
Mixing Times 4 - Avoiding Periodicity
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wrote 3 months ago: A Markov chain is periodic if you can partition the state space such it is possible to be in a parti … more →
What do normal subgroups look like?
wrote 8 months ago: I am in the process of writing a longer post on Galois Theory (see here), and one of the central con … more →
Comparison of group and ring theory
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wrote 1 year ago: Yet again, I’m unable to sleep. Is there any notion (which we could make reasonably precise) o … more →
Lecturer in Mathematics
wrote 1 year ago: Lecturer in Mathematics in Collegiate Education 1. A non-empty set with associative binary operation … more →
Half of the Elements are of Order 2
wrote 1 year ago: Problem? Let be a group of order . Suppose that half of the elements of are of order 2, and the othe … more →
Commutator Subgroup
wrote 1 year ago: Problem? Let be a group and . Let be the smallest subgroup of which contains . (a) Prove that is nor … more →
Index and Order are Relatively Prime
wrote 1 year ago: Problem? If is a normal subgroup in the finite group such that and are relatively prime, show that a … more →
Partitions of a group and normal subgroups
wrote 1 year ago: Let be a group, a normal subgroup of and the set of cosets of Then is a partition of and for all We … more →
Products of Groups (1)
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wrote 2 years ago: Recently, I have been reading some algebra. This has been immensely enjoyable; I had forgotten how m … more →
Review of Group Theory: Interesting Consequence of the First Isomorphism Theorem
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wrote 2 years ago: Point of post: In this post we give one interesting “corollary” (it isn’t actually … more →
A finite group whose order is the product of three distinct primes has a normal Sylow subgroup of largest order
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wrote 2 years ago: Let be a group of order where and , , and are primes. Prove that a Sylow -subgroup of is normal. Rec … more →
Every group of order 36 has a normal Sylow subgroup
wrote 2 years ago: Prove that if is a group of order 36, then has either a normal Sylow 2-subgroup or a normal Sylow 3- … more →
No simple groups of order 144, 525, 2025, or 3159 exist
wrote 2 years ago: Prove that there are no simple groups of order 144, 525, 2025, or 3159. Note that . Let be a simple … more →
Every minimal normal subgroup of a finite solvable group is elementary abelian
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wrote 2 years ago: For any group a minimal normal subgroup is a normal subgroup of such that the only normal subgroups … more →
The Frattini subgroup of a finite group is nilpotent
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wrote 2 years ago: Let be a finite group. Prove that is nilpotent. We will use Frattini’s Argument to show that e … more →
The Frattini subgroup is inclusion-monotone on normal subgroups
wrote 2 years ago: When is a finite group prove that if is normal, then . Give an explicit example when this containmen … more →
Some basic properties of nilpotent groups
wrote 2 years ago: Prove the following for an infinite nilpotent group. Let be a nilpotent group. If is a nontrivial no … more →
Equivalent characterizations of nilpotent and cyclic groups
wrote 2 years ago: If is finite, prove that (1) is nilpotent if and only if it has a normal subgroup of each order divi … more →
For odd primes p, a p-group whose every subgroup is normal is abelian
wrote 2 years ago: Let be an odd prime and let be a -group. Prove that if every subgroup of is normal then is abelian. … more →
