1
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1answer
33 views

Confusion with application of butterfly lemma in Lang's Algebra

In Lemma 3.3 of Serge Lang's Algebra, the so-called Butterfly Lemma is proved: And then Lang proceeds to prove Schreier's Refinement Theorem (highlight mine): In the proof of Schreier's theorem, ...
1
vote
1answer
34 views

Splitting field in multiple field extension

Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$. My problem with this proof is that I do not know ...
2
votes
4answers
83 views

Proper Ideal and Units Proof

Show that an ideal of I R is proper if and only if it does not contain 1 iff and only if it does not contain any units. (1 is the identity element) I'll need to show 3 parts: (1) $\implies$ (2): ...
0
votes
0answers
31 views

Ideal Test Proof

Let $\emptyset \subset I \subseteq R$. Prove that I is an ideal of R if and only if $a-b, ra, ar$, $\in$ $I$ for all $a, b \in I$ and $r \in R$. I know that if I is an ideal in a ring R and $a \in ...
2
votes
1answer
54 views

Integral Domain and no nonzero divisors Proof

Prove that a commutative ring is an integral domain if and only if it has no nonzero zero divisors. I think my main problem is that I'm getting jumbled in the wording! By 'no nonzero zero divisors' ...
0
votes
1answer
29 views

Group Order and Least Common Multiple

Let $G_1,G_2,...G_n$ be groups. Show that the order of an elements $(a_1,a_2,...a_n)$ $\in$ $G_1 \times G_2 \times ... \times G_n$ is lcm($o(a_1),...,o(a_n))$ I know I need to use the fact that the ...
2
votes
1answer
42 views

Centralizer Proof: $A \subseteq C(C(A))$

Show that $ A \subseteq C(C(A))$ Let $G$ be a group and $ A \subseteq G $. The centralizer of a subset of A is the set $C(A)=\{x\in G : ax=xa$ for all $a \in A\}$. *Isn't this obvious because $A=A$ ...
0
votes
1answer
45 views

Group Theory Exponent and Abelian Proof

Let G be a group such that $x, y \in G$ Show that, if $(xy)^2=x^2y^2$ or $(xy)^{-1}=x^{-1}y^{-1}$, then xy=yx. This can also be thought of as the exponent rule $(xy)^n$=$x^ny^n$ if xy=yx is true ...
2
votes
3answers
79 views

$n$-abelian Groups

Show that $(xy)^n=x^ny^n$ if $xy=yx$. I assume I will need 3 different cases: $n < 0$, $n=0$, and $n > 0$. For the $n > 0$ case, can I use induction? For the base case I'll show that ...
0
votes
1answer
31 views

Cardinality of a Monoid and Constant Functions

Let $X$ be a set. Show that $((M(X),\circ)$ has an absorbing element iff $|X|\leq 1$ iff $M(X)$ is commutative. In this problem $((M(X),\circ)$ is a monoid and M(X) is the set of all maps from X to ...
0
votes
2answers
60 views

Absorbing Element is a Unit

Show that an absorbing element of a monoid is a unit if and only if it is the only element. This is an if and only if proof so that means I have to prove it both ways: A implies B and B implies A. ...
1
vote
1answer
37 views

Qustion about field and sub-group

$F$ is a field and and $H$ is finite sub-group of $(F,\cdot)$ ($F$ without the $0_F$). I need to prove that $H$ is cyclic. I can use this fact - Can we conclude that this group is cyclic?. (I don't ...
10
votes
2answers
209 views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
2
votes
0answers
125 views

Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles.

I would like to know if my proof below is correct. Problem Let $p$ be a prime. Show that an element has order $p$ in $S_n$ iff its cycle decomposition is a product of commuting $p$-cycles. Show by ...
1
vote
1answer
58 views

If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some $k\in\mathbb N$.

I would like to know if my proof below is correct. Problem If $\tau = (1,2)(3,4,5)$ determine whether there is a $n$-cycle $\sigma$ ($n\geq 5$) with $\tau = \sigma^k$ for some integer $k$. ...
3
votes
3answers
95 views

If $\Omega = \{1,2,3,\ldots,\}$, then $S_{\Omega}$ is an infinite group.

I would like to know if my proof below is correct. I do not have issues proving that $S_{\Omega}$ is a group; what I am not sure is whether my proof that $\vert S_{\Omega} \vert = \infty$ is correct. ...
0
votes
1answer
31 views

Question about question about finite order elements at $\mathbb{C}^*/U$

At this question - What are element with finite order at $\mathbb{C}^*/U$? I understand that finite order at $\mathbb{C}^*/U$ are only the $e$ elements. Now, I have two questions: It is because ...
1
vote
0answers
58 views

Proving a theorem at groups theory [duplicate]

$G$ is a finite group. Let assume that for every $m\in \mathbb{N}$ there are maximum $m$ elements s.t. $x^m=e\;$. $(x\in G)$ I need to prove that $G$ is cyclic group. I need to use the fact that: ...
3
votes
2answers
53 views

What are element with finite order at $\mathbb{C}^*/U$?

I need to find the elements with finite order at the group - $\mathbb{C}^*/U$. $U$ - is the Circle Uint. $\mathbb{C}^*$ - is $(\mathbb{C}/0,\cdot)$. I need to write also the proof, and I'll be glad ...
2
votes
3answers
71 views

What are the finite order elements of $\mathbb{Q}/\mathbb{Z}$?

I need to find what are the at the group $\mathbb{Q}/\mathbb{Z}$. I think that any element at this group has a finite order, but I don't know how to prove it... I'd like to get help with the proof ...
0
votes
1answer
127 views

How to prove this Inverse Property of Group

I am given a Group $G$ with Projection $*:G\times G \Rightarrow G$ and with these properties: $a*(b*c) = (a*b)*c$ $e*a=a$ $b*a=e$, $b$ is invers Element $a*b=b*a$ I want to prove ...
1
vote
2answers
89 views

Is it ok to prove a subset of a group is an abelian group this way?

I'll admit from the start I'm being lazy, but all the same it makes thing's neater in my opinion - if it's valid. Now it's known that if we have a group $G$ such that $g^2=e,\ \ \forall g \in G$ then ...
1
vote
2answers
75 views

$\left| G:H \right|=2 \implies H$ is normal in $G$

I have a question to the proof of this claim: "Let $G$ be a group. Then $\left| G:H \right|=2 \implies H$ is normal in $G$. " Here is an excerpt of the proof: Let $G= g_1H\biguplus g_2H.$ Since ...
0
votes
1answer
34 views

Question about definition of generator for dihedral groups?

Dummit and Foote mention that a relation for the dihedral group is $rs = sr^{-1}$. Now, I have interpreted the statement to mean $r$ is a rotation of $\frac{2\pi}{n}$ radians and $s$ is an ...
3
votes
1answer
47 views

Proving that a $p$-group operating on a finite set of order not divisible by $p$ has a fixed point.

Let $G$ be a finite group of order $p^e$ for some prime $p$. Let $S$ be a set of size not divisible by $p$. I know that $|G| = |Stab(s)|\cdot|O_s|$ = (stabilizer of $s$)(orbit of $s$) So if there ...
5
votes
1answer
49 views

Is this a valid proof of the contrapositive?

The question is the following: if $a$ and $b$ are distinct group elements, then either $a^2 \neq b^2$ or $a^3 \neq b^3$. I find this difficult to prove directly, so I formulated the contrapositive to ...
3
votes
1answer
58 views

Rephrase the proof “For all odd n, there exists a Group G …” [duplicate]

I am trying to construct a proof and would like to know if I have started it correctly. The proof is as follows. "Prove that for every odd integer n, there is a group with exactly n elements of order ...
1
vote
2answers
575 views

Proof Help: In a group $G$, there exists a $g$ such that $g^2 = e$ [duplicate]

I'm working through an Abstract Algebra book to teach myself, and came across the problem: Prove: If $G$ is a finite group of even order, then there exists a $g\in G$ such that $g^2 = e$ and $g ...
1
vote
2answers
110 views

Let H and K be subgroups of the finite group G and supposes $|H|^{2}>|G|$ and $|K|^{2}>|G|$. Prove $H \cap K$ has at least two elements

So I supposed $|H \cap K|>1$ $\Rightarrow |HK||H \cap K|> |HK|$ Which eventually implied that $\Rightarrow |H \cap K|>|G|$ Thus since G is a group, and H and K are subgroups then the ...
3
votes
1answer
190 views

Finite Abelian groups, G, H, K: $G \times H \cong G\times K$ then $H\cong K$

Let $G,H,$ and $K$ be finite abelian groups then if $G \times H \cong G\times K$ then $H\cong K$. I am trying to use the fundamental theorem for abelian groups to solve this, it is clear intuitively ...
3
votes
2answers
328 views

Prove the third isomorphism theorem

I'm trying to prove the third Isomorphism theorem as stated below Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$ I look up for some ...
2
votes
2answers
54 views

For any normal subgroup $(aN)^n=(a^n) N$ holds

Prove this theorem Let $G$ be a group and $N$ a normal subgroup of $G$. If $a \in G$ and $n \in Z$, then $(aN)^n = (a^n) N$. I know I should prove this theorem in 3 cases where $n = 0$, $n>0$, ...
3
votes
0answers
793 views

Prove that the set Aut(G) of all automorphisms of the group G with the operation of taking the composition is a group

Let $G$ be a group. Say what it means for a map $\varphi: G \rightarrow G$ to be an automorphism. Show that the set-theoretic composition $\varphi \psi = \varphi \circ \psi$ of any two ...
0
votes
2answers
722 views

Prove that any group $G$ of order $p^2$ is abelian, where $p$ is a prime number [duplicate]

Possible Duplicate: Showing non-cyclic group with $p^2$ elements is Abelian "Let $p$ be a prime number. Prove that any group $G$ of order $p^2$ is abelian. You may assume the fact that the ...
1
vote
1answer
50 views

$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute

This is the proof, which I mostly understand except for one bit: You have $h_1 \in H_1$ and $h_2 \in H_2$. We also have $h_1^{-1}(h_2^{-1}h_1h_2) \in H_1$, because $h_2^{-1}h_1h_2 \in ...
4
votes
0answers
147 views

Easy proof of trivial fusion implies normal p-complement

Theorem: Suppose G is a finite group with Sylow p-subgroup P. Then the following are equivalent: The set K of elements of G of order relatively prime to p (the p′-elements) form a subgroup If A and ...
6
votes
1answer
1k views

The Product of Subgroups of an Abelian Group

Reference: Fraleigh p. 58 Question 5.43 in A First Course in Abstract Algebra Let $G$ be an Abelian Group. Suppose $H$ and $K$ are subgroups of $G$, and $HK = \{ xy: x \in H \text{ and } y \in K ...
3
votes
1answer
952 views

Subgroups of Abelian Groups

In the following problems, let $G$ be an Abelian group. 1) Let $H = \{ x \in G: x=y^{2} \text{ for some } y \in G \}$; that is, let $H$ be the set of all the elements of $G$ which have a square root. ...
2
votes
1answer
188 views

Subgroups of Functions

$\mathcal{F}$ is the set of all real-valued functions of a real variable. I'm trying to show that $H$ is a subgroup of $G$. $G = \langle \mathcal{F}, + \rangle, H = \{ f \in ...
1
vote
1answer
220 views

Determining Subgroups

Could I get some feedback on my work below? Thanks in advance. $G = \langle \mathbb{R}, + \rangle , H = \{ x \in \mathbb{R}: \tan x \in \mathbb{Q} \}$ (i). If $\tan a$ and $\tan b$ $\in H$, then ...
2
votes
2answers
268 views

Powers and Roots of Group Elements

Let $G$ be a group, and $a$, $b \in G$ $(bab^{-1})^{n} = ba^{n}b^{-1}$, for every positive integer $n$ \begin{align*} \text{Let P(n) be the statement: } (bab^{-1})^{n} &= ba^{n}b^{-1} \newline ...
11
votes
5answers
556 views

Finite Abelian Group

Let $G$ be a finite abelian group, $G = \{e, a_{1}, a_{2}, ..., a_{n} \}$. Prove that $(a_{1}a_{2}\cdot \cdot \cdot a_{n})^{2}$ = $e$. I've been stuck on this problem for quite some time. Could ...
2
votes
1answer
432 views

Counting Elements and Their Inverses

The problem I am attempting to prove is the following: In any finite group $G$, the number of elements not equal to their own inverse is an even number. Caveat: I have had very limited experience ...