Purchase Solution

Subgroup proof

Not what you're looking for?

Ask Custom Question

If |G|=p^n, where p is a prime, show that G has subgroups G_0, G_1, . . ., G_n with 1= G_0 <= G_1 <= . . . <= G_n =G such that [G_i : G_{i-1}] = p, 1 <= i <= n.

I need a rigorous, detailed proof of this to study please.

Purchase this Solution
Solution Summary

This provides an example of proving given subgroups.

Solution Preview

Proof:
We know that if |G| is divisible by p, then G has a subgroup of order p.
Now |G| = p^n. Let G_n = G.
Since |G| ...

Purchase this Solution
Free BrainMass Quizzes
Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Probability Quiz

Some questions on probability

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

View More Free Quizzes
Related BrainMass Solutions

  • Sylow p-Subgroups, Conjugacy and Abelian Groups

    Proof: From the condition, we know that is a sylow-p subgroup of , is a normal subgroup of , then is a p-subgroup of .

  • Subgroup proofsno

    Therefore, is a subgroup of . Hence is isomorphic to a subgroup in . (b) Proof: If has a subgroup with finite index , since is simple and from the result in (b), is isomorphic to a subgroup of .

  • Subgroups

    So is the smallest subgroup of G that contains X. A proof involving subgroups is provided. The proof is concise.

  • Groups and matrices

    V= four group I really dont know how to do these, I need to learn them in written proof form and I struggle. attached This provides an example of a proof regarding a matrix with infinite order, and a proof regarding normal subgroups.

  • Normal Subgroups

    So if and only if is abelian A normal subgroup is investigated. The proof is elegant.

  • Groups and proofs

    Problem #2 Proof: is a subgroup of a group and we know that . I want to show that . We note, for any two right cosets and in , either or . This means any two right cosets are either equal or disjoint.

  • Proof regarding direct sum of 2 copies of (Q,+)

    223661 Proof regarding direct sum of 2 copies of (Q,+) Use the fact that any finitely generated subgroup of (Q,+), (the rationals under addition), is cyclic to show that (Q,+)+(Q,+), (the direct sum of two copies of (Q,+)), is not isomorphic to (Q,+).

  • Subgroup proofs

    This provides an example of completing a proof about the largest normal subgroup in a group and Sylow 5-subgroups.

  • Normal subgroup proof

    107391 Normal subgroup proof Let G be a finite group, let N be a normal subgroup of G, and let x be an element of G. Show that if the order of x in G is relatively prime to |G|/|N|, then x is an element of N.

View More Related Solutions