Abstract Algebra

Prove that an integer $a$ is invertible modulo $n$ if and only if $\gcd(a, n) = 1$.

Consider the groups $G = \mathbb{R}$ under addition and $H = \mathbb{R}^+$ under multiplication. Show that the function $\Phi: G \to H$ defined by $\Phi(x) = e^x$ is a group homomorphism.

Suppose we have a set of 2 by 2 matrices whose determinants are non-zero. Consider the subset of these matrices whose determinant is 1. Using the two-step subgroup test, verify if this subset is closed under multiplication and if an element belongs to this subset, confirm that its inverse also belongs to this subset.

Given a finite subset of a group, verify if the subset is closed under the group operation to determine if it is a subgroup.

Let $R$ be an integral domain. Suppose there exists a nonzero $y$ such that $y + y + \ldots + y$ (n times) = 0, where $n$ is an integer greater than 1. Show that $x + x + \ldots + x$ (n times) = 0 for all $x$ in $R$.

Given a group with order 432,000 and its decomposition as $C_{60} \times C_{60} \times C_{24} \times C_{5}$, determine whether these numbers represent invariant factors or elementary divisors. Then, find the correct representation of invariant factors.

Recall that if $R$ is a PID and $M$ is a finitely generated $R$-module, then $M$ is isomorphic to the direct sum of cyclic modules whose annihilators are generated by powers of primes in $R$.

Convert the invariant factor decomposition of a module into the elementary divisor decomposition by factoring each invariant factor into a product of prime powers and using the Chinese Remainder Theorem for modules.

Assume $F$ contains all eigenvalues of $T$, and demonstrate that each invariant factor can be factored into powers of linear polynomials where the linear factors correspond to eigenvalues of $T$.

Explain how the group U(8) of units modulo 8 (under multiplication modulo 8) is isomorphic to the Klein 4 group $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

Describe the two isomorphism classes for groups of size four, particularly focusing on $\\\mathbb{Z}/4\\\mathbb{Z}$ and the Klein 4 group, and provide an example group for each.

Determine if S3 is isomorphic to a cyclic group of order 6 and explain why or why not.

Determine if the integers under addition are isomorphic to the even integers under addition, and provide reasoning for your answer.

Consider a homomorphism $\ :$$\ :$from the additive group of all real-valued functions to the set of real numbers, given by the definition \(f) = f(0) . Determine the kernel of this homomorphism.

Given a homomorphism $\phi: \mathbb{Z} \to \mathbb{Z}_8$ defined by $\phi(x) = x \mod 8$, determine if it preserves the homomorphism property.

Given a homomorphism $\phi: GL(2, \mathbb{R}) \to \mathbb{R}^*$ defined by $\phi(A) = \det(A)$, show that it satisfies the homomorphism property.

Find the kernel of the homomorphism $\phi: \mathbb{R}^* \to \mathbb{R}^*$ defined by $\phi(x) = x^2$.

Compute $(45 + 38) \mod 7$ by modding first.

Compute $(45 \times 38) \mod 7$ by modding first.

Prove that $(a \cdot b) \mod n = (a' \cdot b') \mod n$ when $a \mod n = a'$ and $b \mod n = b'$.