Subgroups of Prime Order: If G has no nontrivial subgroups, then it must have prime order.

If G has no nontrivial subgroups, then it must have prime order.

Normalizer of a Group or Centralizer of a Group: If aєG define N(a) = {xєG|xa = ax}.Show that N(a) is a subgroup of G. N(a) is usually called the Normalizer or Centralizer of a in G.

If aєG define N(a) = {xєG|xa = ax}. Show that N(a) is a subgroup of G. N(a) is usually called the Normalizer or Centralizer of a in G.

Centre of a Group: If G is a group, the centre of G, Z is defined by Z = {zєG|zx = xz, all xєG}. Prove that Z is a subgroup of G.Or Prove that Z is a normal subgroup of G.

Modern Algebra Group Theory (XXX) Subgroups of a Group Centre of a Group If G is a group, the centre of G, Z is defined by Z = {zєG|zx = xz, all x&# ...continues

Cyclic Groups: Prove that any subgroup of a cyclic group is itself a cyclic group.

Modern Algebra Group Theory (XXXI) Subgroups of a Group Cyclic Groups Prove that any subgroup of a cyclic group is itself a cyclic group.

The Order of an Element of a group: If aєG, a^m = e prove that O(a)|m.

Modern Algebra Group Theory (XXXII) Subgroups of a Group The Order of an Element of a group If aєG, a^m = e prove that O(a)|m.

The Order of an Element of a group: If in the group G, a^5 = e, aba^(-1) = b^2 for a,bєG find O(b).

Modern Algebra Group Theory (XXXIII) Subgroups of a Group The Order of an Element of a group If in the group G, a^5 = e, aba^(-1) = b^2 for a,bєG find ...continues

Cosets of Subgroups of a Group and Normal Subgroups of a Group: If H is a subgroup of G such that the product of two right cosets of H in G is again a right coset of H in G, prove that H is normal in G.

Modern Algebra Group Theory (XXXIV) Cosets of Subgroups of a Group Normal Subgroups of a Group If H is a subgroup of G such that the product o ...continues

Group Theory : Subgroups and Cosets - A subgroup N of G is a normal subgroup of G if and only if the product of two right cosets of N in G is again a right coset of N in G.

A subgroup N of G is a normal subgroup of G if and only if the product of two right cosets of N in G is again a right coset of N in G.

Group Theory : If G is a group and H is a subgroup of index 2 in G, prove that H is a normal subgroup of G.

If G is a group and H is a subgroup of index 2 in G, prove that H is a normal subgroup of G.

Group Theory : If N is a normal subgroup of G and H is any subgroup of G, prove that NH is a subgroup of G.

If N is a normal subgroup of G and H is any subgroup of G, prove that NH is a subgroup of G.