Prove that Z[√2]/(3+√2)\mathbb{Z}[\sqrt 2]/(3+\sqrt 2) isomorphic to Z7\mathbb{Z}_7 [closed]

Closed. This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for Mathematics Stack Exchange. Closed 2 years ago. Improve this question I am really stuck here, could you please give an insight or show the proof step by … Read more

Confused by quotient group (whats the operation): Show quotient group GLn(K)/SLn(K)GL_n(K)/SL_n(K) is abelian.

In my introductory abstract algebra course, the quotient group G/H was defined as G/H={gH:g∈G} which is a set of sets. In an exercise, I should show that for the group of invertible matrices GLn(K) over a field K and the normal subgroup SLn(K) the quotient group is abelian. I’m horribly confused. What is the operation … Read more

The authors of algebra books don’t write this group H′H’ in their books. Why?

Let G be a group. Let N be a normal subgroup of G. Let H be a subgroup of G such that N⊈. Define a relation \sim on H as follows: a\sim b if and only if a^{-1}b\in N. Then, this relation is an equivalence relation on H. Let \overline{a}:=\{x\in H\mid a\sim x\} for a\in … Read more

Isomorphism class of quotient group

I am dealing with a problem of Algebraic Topology and I have reached to this: I have a group G (abelian) and I have to find the possible isomorphism classes of G, if : G/\mathbb Z_2 \simeq \mathbb Z_2 Now I am not sure how to proceed. Considering that G is abelian and \mathbb Z_2 … Read more

Is SU(n)/ZnSU(n)/\mathbb Z_n a Lie group?

I know that Zn is the center of the Lie group SU(n). Therefore, the quotient SU(n)/Zn forms a group. However, I am not sure whether it also forms a Lie group. I also wonder what is su(n)/Zn at the Lie algebra level. So far, I have only seen examples of quotient group G/H in which … Read more

Consider the group $\mathbb{Z}_{20}$

Consider the group $\mathbb{Z}_{20}$ and let $H = \langle [4]\rangle $ be the subgroup generated by $4$. List all the elements of $\mathbb{Z}_{20}/H$ and show that the quotient is cyclic. I think I just have a lack of understanding of terminology here. I know that $\mathbb{Z}_{20}/H$ just represents the left cosets of $H$, but since … Read more

G/N=1G/N = 1 iff N=GN = G and G/N=GG/N = G iff N=1N = 1

Suppose that G is a finite group and N⊴. Then G/N \cong \{1_G\} \iff |G/N| = |\{1_G\}| \iff \frac{|G|}{|N|}= 1 \iff |G| = |N| \iff N=G \text, which answers this question. Likewise, G/N \cong G if and only if N is trivial. I thought these facts were true for infinite groups as well, but now … Read more

Let $f(x) =x^3+x+1 \in \mathbb{Q} [x] $. If ideal $I=(f(x)) $, find the inverse of $x^2+x+1$ in quotient $\mathbb{Q} [x] /I$.

Let $f(x) =x^3+x+1 \in \mathbb{Q} [x] $. If ideal $I=(f(x)) $, find the inverse of $x^2+x+1$ in quotient $\mathbb{Q} [x] /I$. I am having trouble with this. What exactly is the quotient $\mathbb{Q} [x] /I$ here? I guess it’s polynomials with degree 2 or less which in multiplication satisfy the relation $ x^3+x+1=0$. How do … Read more

If $E/F$ is a Galois extension with abelian Galois group, then $E$ is a tower of quadratic extension iff $[E:F]$ is a power of $2$

I am trying to understand the proof of: If $E/F$ is a Galois extension with abelian Galois group, then $E$ is a tower of quadratic extension iff $[E:F]$ is a power of $2$. It is the proof above. But I cannot understand why we must use quotient here. I think even the subgroup of order … Read more

If GG is a non-abelian finite group, then |Z(G)|≤14|G||Z(G)| \leq \frac {1}{4} |G|

I know this is question has been asked several times on here, only hints given ,but just want to check if I have the right idea. My attempt: Suppose G is non abelian finite group and |Z(G)|>14|G|. Since Z(G) is a subgroup of G then it’s order must divide that of G , i.e it … Read more