Let H={1,17,41,49,73,89,97,113} under multiplication modulo...

Let $H=\{1,17,41,49,73,89,97,113\}$ under multiplication modulo 120 . Write $H$ as a external direct product of groups of the form $Z_{2} k .$ Write $H$ as an internal direct product of nontrivial subgroups.

Key Concepts

Group Theory

Group theory is the branch of abstract algebra that studies the algebraic structures known as groups, which consist of a set equipped with an operation that combines any two elements to form a third element while fulfilling the properties of closure, associativity, having an identity element, and every element possessing an inverse. It provides the framework for understanding symmetry and structure in various mathematical systems.

Multiplicative Group of Integers Modulo n

A multiplicative group modulo n is formed by the set of integers that are relatively prime to n, under the operation of multiplication modulo n. This group structure is crucial in number theory and cryptography, and it serves as a common example in the study of finite groups when considering modular arithmetic operations.

Cyclic Groups

A cyclic group is a group that can be generated by a single element, meaning every element in the group is some power (or multiple, in the additive case) of this generator. Their simple structure makes them a fundamental example in group theory and a key building block when analyzing more complex groups.

Direct Product of Groups

The direct product of groups is a method of constructing a new group from two or more given groups, where the set of the new group consists of ordered tuples from the original groups, and the group operation is performed component-wise. This concept is central in understanding how larger groups can be decomposed into simpler, well-understood subgroups.

External Direct Product

The external direct product is a construction technique where independent groups are combined to form a larger group in which the operation is defined component-wise. It emphasizes the separate origins of each component group, and the resulting group is structurally equivalent to having each subgroup operate independently within a combined framework.

Internal Direct Product

An internal direct product refers to a situation within a given group where the group can be expressed as a product of its subgroups. This requires that the subgroups have trivial intersections (i.e., they intersect only at the identity) and that every element of the group can be uniquely written as a product of elements from these subgroups. It provides an intrinsic way of decomposing the group into simpler parts.

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