CH 2 - Subgroups

Question 1

Define a subgroup H of G and give 2 further equivalent characterisations of a subgroup.

Answer 1

Question 2

Prove the alternating group is a subgroup of the symmetric group.

State the theorem on the intersection of subgroups.

Answer 2

Question 3

Define the subgroup generated by X.

Define it in 2 further equivalent ways.

Answer 3

Question 4

Define the subgroup generated by a single element.

Define a cyclic group.

Define a generator of G.

Define the cyclic subgroup generated by x.

Answer 4

Question 5

What can we say about the size of the subgroup generated by x and the order of x?

Define < X > for a finite group G.

State the Generation Algorithm for finding all elements of < X > for a finite group G.

Answer 5

Question 6

Define the Dihedral Group of order 2n.

State four properties of the Dihedral Group D_2n.

Answer 6

Question 7

Define the right coset of H
with representative x.

Answer 7

Question 8

Define the index of H in G.

State Lagrange’s Theorem.

State a corollary of Lagrange’s Theorem about the order of x and G.

State a corollary on G of prime order p.

Answer 8