# mt1 - H ⊆ G be a subgroup of order 24 Suppose that there...

This preview shows pages 1–6. Sign up to view the full content.

C Math 250A PROFESSOR KENNETH A. RIBET First Midterm Examination September 30, 2004 12:40–2:00 PM Name: Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Explain your answers in full English sentences as is customary and appropriate. Your paper is your ambassador when it is graded. Problem: Your score: Total points 1 6 points 2 6 points 3 6 points 4 6 points 5 6 points Total: 30 points 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
1. If n is a positive integer, ﬁnd an m 1 so that the alternating group A m contains a subgroup isomorphic to the symmetric group S n . 2
2. Prove that every group of order 312 = 2 3 · 3 · 13 has a proper non-trivial normal subgroup. 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
3. Let G be a group of order 120, and let

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: H ⊆ G be a subgroup of order 24. Suppose that there is at least one left coset of H in G (other than H itself) that is also a right coset of H in G . Prove that H is a normal subgroup of G . 4 4. Let G be a group and let H be a subgroup of G such that the index ( G : H ) is ﬁnite. Prove that there is a normal subgroup H of G such that H ⊆ H and such that ( G : H ) is ﬁnite. Show further that there is an n ≥ 1 so that g n ∈ H for all g ∈ G . 5 5. Let G be a ﬁnite group, and let H be a normal subgroup of G . Let P be a p-Sylow subgroup of H , and let K be the normalizer of P in G . Establish the equality G = HK . 6...
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

mt1 - H ⊆ G be a subgroup of order 24 Suppose that there...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online