Let H be a subgroup of a group G . Prove that the set H Z(G)= {h z |h ∈H, z ∈Z(G)} is a subgroup

step 1 So, if x1, x2 belongs to k, then there are, there are n1, comma, n2 belonging to n, such that x

Let H be a subgroup of a group G . Prove that the set H...

Key Concepts

Subgroup

A subgroup is a subset of a group that is itself a group under the same binary operation. To qualify as a subgroup, the subset must contain the identity element of the group, be closed under the group operation, and include the inverses of its elements. This concept is fundamental in group theory as it allows for the study of smaller, well-behaved pieces within a larger algebraic structure.

Center of a Group

The center of a group, denoted as Z(G), is the set of all elements that commute with every element of the group. This means that for any element z in Z(G) and any element g in G, the relation zg = gz holds. The center is an important concept because it always forms a subgroup of G and frequently plays a role in simplifying problems by reducing non-commutative elements.

Normal Subgroup

A normal subgroup is a subgroup that is invariant under conjugation by any element of the group. In other words, if N is a normal subgroup of G, then for every element g in G and every n in N, the element gng?¹ is still in N. This property is crucial because the center of a group is always a normal subgroup, which permits the formation and manipulation of quotient groups and facilitates reasoning about subgroup products.

Product of Subgroups

The product of subgroups involves forming a new set by taking all possible products of elements from two subgroups. In the specific case where one of the subgroups is normal, the product of the two subgroups (i.e., {hz | h in H, z in Z(G)}) is itself a subgroup. This result stems from the normality ensuring that the subgroup elements commute in a way that preserves the subgroup structure when combined.

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