# Tagged Questions

20 views

### $|G| \geq 3$ is finite and has subgroup with index more than 4 then $G$ is not simple.

suppose $G$ is a finite group with more than three elements,prove that if $H$ will be a proper subgroup of $G$ with this property that $[G:H] \geq 4$ then $G$ is not simple. I found this question a ...
55 views

### Isomorphisms in finite abelian groups

Let G be an abelian group of order 175 (=5*5*7). Assume $x^5=e$ has at least seven solutions. What is G isomorphic to? I see and can show that G is isomorphic to the its Sylow subgroups (orders 7 and ...
21 views

22 views

### Sylow subgroups and normalizers [duplicate]

Let $H$ be a Sylow 3-subgroup and $K$ a Sylow 5-subgroup of a finite group $G$. Suppose $|H|=3$ and $|K|=5$ and $N_G(K)$, has an element of order 3. Show that $N_G(H)$ has an element of order 5.
37 views

41 views

### Property of isomorphic subgroups in finite groups

I have the following question: Does there exist a finite group $G$ and two subgroups $U,H\leq G$, s.t. the following properties are satisfied: a) $H\cong U$. b) There is no subgroup $L$, s.t. ...
29 views

### $2$-Sylow subgroups of $S_{14}$

I need to describe what are the $2$-Sylow subgroups of $S_{14}$. I don't know how to start thinking about this since is it too large. I know that all that I must do is to find one of them since all ...
14 views

### Show that the orbit of $H$ containing $x$ is equal to the right coset $Hx$

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $X$ be the set of elements of $G$. Let $\ast : H \times X \to X$ be given by $$h \ast x = hx (h \in H, x \in X)$$. QUESTION: Let $x \in X$. ...
87 views

73 views

### Can someone please explain the word problem (from group theory) in Calculus III layman's terms

I'd appreciate it if someone could offer a schematic explanation of the word problem from group theory in terms which someone at a calculus III level (but without any background in group theory or ...
195 views

### The use of conjugacy class and centralizer?

This is more or less for a conceptual and better-understanding question in group theory and in representation theory: (1) Why are conjugacy class and centralizer important concepts in the group / ...
117 views

### Is $\langle a,b\; |\;a^7 = 1, ab = b^3a^3\rangle$ finite?

I've been playing a little with group definitions to see what kind of things I can make up. I'm struggling to prove that this group is finite. Can anyone point me in the right direction?
56 views

### For a finite group $G$, if $N, G/N$ have relatively prime orders with $N$ normal, then for any automorphism on $g$, $g(N) =N$.

For a finite group $G$, if $N, G/N$ have relatively prime orders with $N$ normal, then for any automorphism $g$, $g(N) =N$. I can prove this for the case when there is a subgroup $H$ with the same ...
40 views

### Does G necessarily have a subgroup H…

I'm confused on an abstract math question. Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup H such that $H≅G/K$ and $H∩K=⟨0⟩$. I think it is ...
56 views

### How to show that a group is finite and also normal

Let $G$ be an finite group and $H$ normal subgroup of $G$. Show $\left|G\big/H\right|=\left|G\right|$ if and only if $H=\{e\}$. Firstly I do not know how to show that $G$ is finite. Next I know that ...
218 views

### Can we uniquely determine a group given the orders of its elements?

Given a finite group $G$ and its order, consider a scenario in which we also know the orders of each of its elements. Does this information alone uniquely determine the group? If not, can we at ...
37 views

### A detail in Baer Theorem

I refer to Martin Isaac book, "Finite Group Theory", Theorem 2.12 pag. 55. In the proof there is a detail I can't understand. Our hypotesis are the following: $G$ finite group, $H\leq G$ s.t. ...
108 views

### Is the group $(G,*)$ abelian?

Let $(G,*)$ be a finite group in which the sets $C_a$={$x\epsilon G$|$ax=xa$} have the same cardinality, for all $a \epsilon G$ \{e}. My question is: is the group abelian?
73 views

### Using the Fundamental Theorem of Finite Abelian Groups

Let $G$ be a finite abelian group and let $p$ be a prime that divides the order of $G$. Use the Fundamental Theorem of Finite Abelian Groups to show that $G$ contains an element of order $p$. The ...
94 views

### For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?

Let $(G,\ast)$ be a group with identity $e$ and cardinality $2n$ for some $n\in\omega$. Then, does there exist $x\in G$ such that $x\ast x=e$ and $x\neq e$?
44 views

### Using that $G$ is isometric to a subgroup of $S_G$ to prove something about $G$

I am doing the following exercise for an assignment: Assume that $G$ is any finite group with non-trivial elements such that $bab^{-1}=a^{-1}$. Let $k$ be a natural number and use induction to ...
20 views

### Calculate the order of the generalized quaternion group and list its elements

Let $n \in \mathbb N$ and $w \in G_{2^n}$ a primitive $2^n$-th primitive root. Consider the matrices $R=\begin{pmatrix}\omega & 0\\0&\omega^{-1}\end{pmatrix}$ and ...
99 views

### Explicitly computing the isomorphism class of the tensor product of two finite abelian groups

How do I compute the isomorphism class of $A\otimes_\mathbb{Z} B$, where $A$ and $B$ are abelian of finite order? I can do this for a few examples, but I am unsure of how to proceed in the ...
46 views

### About product of $k$ commutators

I need help in folloving lemma. Lemma: Any product of $k$ commutators is expressible in the form $a_1^{-1}a_2^{-1}...a_{2k}^{-1}a_1a_2...a_{2k}$ I guess author means that any product ...
124 views

### Are there homomorphisms of group algebras that don't come from a group homomorphism?

Given a finite group $G$, one can define the group algebra $\mathbb{C}[G]$ as the algebra having the elements of $G$ as a basis, with the multiplication of $G$. Clearly, any group homomorphism induces ...
49 views

### Groups and subgroups

I have been told that {0, 2, 4, 6, 8} is a subgroup of the multiplicative integer mod 10. I know that the operation is multiplication, so I understand that every element has its inverse within the ...
68 views

### Show that the p-Sylow subgroup is normal in $G$

Let $G$ be a finite group and suppose that $\phi$ is an automorphism of $G$ such that $\phi^3$ is the identity automorphism. Suppose further that $\phi(x) = x$ implies that $x = e$. Prove that for ...
### Prove $G$ has a normal Sylow subgroup
Let $|G|=pqr$ where $p, q$ and $r$ are prime and $p < q < r$. Prove that $G$ has a normal Sylow subgroup for either $p, q$ or $r$. Let $n_p, n_q, n_r$ denote the number of Sylow subgroups for ...
### irreducible polynomial of $\alpha$ over $\mathbb{Q}$
Let $\epsilon=\exp({\frac{2i\pi}{9}})$ and $\mathbb{Q}(\epsilon)$ be a cyclotomic extension of $\mathbb{Q}$ of order 9. If $\mathbb{Q}(\alpha)$ is a subfield of $\mathbb{Q}(\epsilon)$, determine the ...