Let p be a prime and d a divisor of p-1. Show that the d th...

Let $p$ be a prime and $d$ a divisor of $p-1$. Show that the $d$ th powers form a subgroup of $U(Z / p \mathbb{Z})$ of order $(p-1) / d$. Calculate this subgroup for $p=11, d=5 ; p=17$, $d=4 ; p=19, d=6$

Key Concepts

Primitive Roots

A primitive root modulo p is a generator of the cyclic group U(Z/pZ). This concept is essential because, with a primitive root g, every element of U(Z/pZ) can be described as g^k for some integer k. Calculating dth power subgroups usually involves computing g^(dk) modulo p, which directly leverages the idea of a primitive root to yield the subgroup elements.

Group of Units modulo a Prime

When working modulo a prime p, the set of integers from 1 to p?1 that are relatively prime to p forms a group under multiplication, often denoted U(Z/pZ). This group is fundamental in number theory because it has nice algebraic properties, including being cyclic, which simplifies many computations and proofs related to modular arithmetic.

Cyclic Groups

A group is cyclic if there exists an element such that every member of the group can be written as a power of that element. In the context of U(Z/pZ), the cyclic structure means that for some primitive root g, every element can be expressed as g raised to some exponent. This property is crucial since it allows us to easily analyze subsets defined by taking powers, such as the set of dth powers.

Subgroups Formed by dth Powers

In a cyclic group, the set of all dth powers forms a subgroup because taking dth powers is a group homomorphism. The image of this homomorphism is a subgroup, and when d divides the group order, the subgroup of dth powers has order (p?1)/d. This concept illustrates how exponentiation can be used to partition cyclic groups into meaningful subgroups.

Order of Subgroups and Lagrange’s Theorem

Lagrange’s Theorem states that the order of any subgroup of a finite group divides the order of the group. In this context, since the subgroup of dth powers is formed by the image of a homomorphism on a cyclic group of order (p?1), its order must divide (p?1). The result that its order is (p?1)/d is a direct application of this theorem, revealing the structure and size of these power-based subgroups.

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