Centers of groups and products of groups

  • Thread starter hgj
  • Start date
  • Tags
    Groups
In summary, to prove that the center of the product of two groups is the product of their centers, one must define the center of a group as the set of elements that commute with all other elements in the group. For the product of two groups, G and H, the center would be defined as GxH={(g,h) such that g in G and h in H}. To further understand this concept, one could consider the product of the centers of G and H, which would be defined as P={z in G such that zg=gz for g in G} and Q={z in H such that zh=hz for all h in H}. With this understanding, one could then define the center of GxH,
  • #1
hgj
15
0
I need to prove that the center of the product of two groups is the product of their centers.

If I let G and H be two groups, then from definitions, the center of G is Z(G)={z in G such that zg=gz for g in G} and the center of H is Z(H)={z in H sucht that zh=hz for all h in H}. Also, the product of G and H is GxH={(g,h) such that g in G and h in H}. My problem right now is that I'm not sure how to define the center of GxH and I'm not sure how to define the product of Z(G) and Z(H). I'm hoping that if I could understand these two things, I could do the problem.
 
Physics news on Phys.org
  • #2
GxH is a group. Call it K. Do you know how to define Z(K)?

Z(G) and Z(H) are groups. Call them P and Q. Do you know how to define PxQ?
 

1. What is a center of a group?

The center of a group is the set of elements that commute with every other element in the group. In other words, it is the set of elements that do not change the group's structure when multiplied by any other element in the group.

2. How is the center of a group related to the group's commutator subgroup?

The center of a group is a subgroup of the group's commutator subgroup. This means that the center is a subset of the commutator subgroup and also satisfies all the properties of a subgroup, such as closure under multiplication and inverses.

3. How is the center of a group related to the group's normal subgroups?

If a group's center is non-trivial (contains more than just the identity element), it is always a normal subgroup of the group. This means that the center is invariant under conjugation by any element in the group.

4. What is the product of groups?

The product of groups is a new group formed by taking the Cartesian product of the two original groups. This means that the elements of the product group are ordered pairs, where the first element comes from the first group and the second element comes from the second group. The group operation is defined component-wise, meaning that the product of two ordered pairs is also an ordered pair, with each component being the product of the corresponding components of the original ordered pairs.

5. How is the center of the product of two groups related to the centers of the individual groups?

The center of the product of two groups is the product of the centers of the individual groups. In other words, the elements of the center of the product group are ordered pairs, where the first element is an element of the center of the first group and the second element is an element of the center of the second group.

Similar threads

Lagrange multipliers understanding
    • Calculus and Beyond Homework Help
Replies
8
Views
466
Do Isomorphic Groups Have Isomorphic Centers?
    • Calculus and Beyond Homework Help
Replies
3
Views
995
Understanding Schur's First Lemma in Group Representations
    • Calculus and Beyond Homework Help
Replies
2
Views
734
Compute the residue of a function
    • Calculus and Beyond Homework Help
Replies
2
Views
865
Commutator group in the center of a group
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
How to show that this expression with tensors reduces to zero?
    • Calculus and Beyond Homework Help
Replies
6
Views
556
Internal Direct Product
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
Conjugacy and Stabilizers in Group Actions
    • Calculus and Beyond Homework Help
Replies
1
Views
763
Is the Direct Product of Groups Associative and Have an Identity Element?
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Characterizing subgroups of a cyclic group
    • Calculus and Beyond Homework Help
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top