2
votes
1answer
35 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
1
vote
1answer
51 views

every Abelian group is a converse lagrange theorem group

Let $G$ be a finite abelian group, then $G$ has a subgroup of order $n$ if and only if $n\mid G$. Proof: by Lagrange if $H\leq G$ then $|H|$ divides $|G|$ so this proves one of the implications. We ...
3
votes
1answer
40 views

Groups - Prove that every element equals inverse of inverse of element

This is my first proof about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. We use $^{-1}$ to denote inverse ...
2
votes
1answer
25 views

Multiplication between a normal subgroup and an arbitrary subgroup.

Given $G$ a group. $N$ a normal subgroup of $G$, and $H$ an arbitrary subgroup of $G$. Prove that $G=NH$ is a subgroup of $G$. I have to prove that $NH=HN$. But for every $h\in H$ we have that ...
1
vote
1answer
47 views

Exercise on characterization of free abelian groups

I was wondering if someone can please check my work on a homework problem. This is from the graduate Hungerford text. Chapter 2.1, number 3. Let $X=\{a_i\ |\ i\in I\}$ be a set. Then the free abelian ...
5
votes
2answers
57 views

Subgroups of order $5$ in Icosahedral group

Let $G$ denote the orientation preserving isometries of Icosahedron. I want to show the following using group-theoretic notions: Let $N\leq G$ be a subgroup with order $5.$ Show that it is a ...
3
votes
2answers
50 views

Group theory exercise - verification?

I'm self-studying abstract algebra, and this is the first non-trivial group theory exercise I've done. Although it's a well-known result, I'd like to make sure it is correct as it took a good few ...
1
vote
2answers
83 views

Show that $Z\times Z$ is not cyclic… [duplicate]

The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra. Recall that an infinite cyclic group is isomorphic to ...
2
votes
1answer
42 views

can a group with non-trivial center be isomorphic to a normal subgroup of its group of automorphisms?

i think (tho would be grateful for error-check) that the line of reasoning below suggests any group with trivial center is isomorphic to a normal subgroup of its automorphism group. question does the ...
2
votes
1answer
38 views

Please check my proof on: $\sim$ is an equivalence relation $\Leftrightarrow S<G$

Problem: Let $\emptyset\ne S\subset G$, where $G$ is a group, and define a relation on $G$ by $a\sim b\Leftrightarrow ab^{-1}\in S$. Show that $\sim$ is an equivalence relation if and only if $S$ is a ...
1
vote
1answer
68 views

Artin Algebra 2.8.3 “Does every group whose order is a power of a prime $p$ contains an element of order $p$?”

I'm not sure whether or not my answer and proof for this question are valid. Could you point out any flaw? Let $G$ be an arbitrary group, an arbitrary element of $G$ be $g$ and $|G|=p^n$. Since a ...
2
votes
1answer
81 views

Subgroup contained in all other subgroups

This is Problem 2.13.10 from Herstein, Topics in Algebra: Let $G$ be a finite abelian group such that it contains a subgroup $H_0 \neq (e)$ which lies in every subgroup $H\neq (e)$. Prove that $G$ ...
7
votes
4answers
94 views

Herstein Question: $G^{i}$ normal in $G$?

I just wanted to ask a quick question. I'm going over the second edition of I.N. Herstein's topics in algebra and one of his exercises asks the reader to prove that each $G^{i} $ is a normal subgroup ...
1
vote
0answers
36 views

“Bypass” Operations and Groups

So I recently stumbled on this (pdf) collection of group theory related Putnam problems. Problem 1978 A-4 defines a "bypass" operation to be a mapping $\circ:S\times S\mapsto S$ such that $$(w\circ ...
4
votes
2answers
83 views

Error in Hungerford's algebra proof? Left id & inv = group

Prop 1.3 in Hungerford's Algebra said that if $G$ is a semigroup and there exist a left identity and each element have a left inverse, then $G$ is a group. The proof (and in fact, even the proposition ...
0
votes
1answer
26 views

Direct Product of Torsion Subgroups

So I came up with this theorem while studying, and concocted a small proof, and I was wondering if someone could verify it, as I am very new to torsion groups/elements. I am open to all criticism. ...
1
vote
1answer
39 views

Proof of coset and normal subgroup

I have this question: Let $G$ be a group, $a,b\in G$ and let $H$ be a subgroup of $G$. i) Give the definition of the coset $aH$ ii) Prove that $aH = bH$ if and only if $a^{-1}b\in H$ ...
2
votes
0answers
39 views

Counter example of monotone union [duplicate]

I saw this exercise in "Elements of Abstract and Linear Algebra" by E. H. Connell: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, ...
2
votes
3answers
70 views

Monotone Union of subgroups being subgroup

I saw this exercise in a book: Suppose $G$ is a group. Suppose $T$ is an index set and for each $t \in T$, $H_t$ is a subgroup of $G$. Furthermore, if $\{H_t\}$ is a monotonic collection, show that ...
2
votes
1answer
47 views

If a group doesn't have subgroups of index 2 and 3, then any subgroup of index 4 is normal.

Let $G$ be this group and $H$ be any subgroup of index 4. $G$ acts on the set of left cosets of $H$ in $G$, which is a homomorphism $\varphi: G\to Aut(G/H) = S_4$. It is easy to see that $\ker ...
1
vote
1answer
42 views

Orbit-Stabiliser Theorem Application

Question Let $G$ be the symmetric group $S_n$ acting on the $n$ points $\lbrace 1, 2, 3, . . . , n\rbrace$, let $g\in S_n$ be the n-cycle $(1,2,3,. . . , n)$. By applying the Orbit-Stabiliser Theorem ...
4
votes
4answers
303 views

Prove that if a group contains exactly one element of order 2, then that element is in the center of the group.

I'm stuck at this question. Can someone please help me? Prove that if a group contains exactly one element of order 2, then that element is in the center of the group. Let $x$ be the element of ...
12
votes
0answers
107 views

Let $f$ and $g$ be two group homomorphisms from $G$ to $G'$. Is $H$ a subgroup of $G$?

Can someone please verify this? Let $f$ and $g$ be two group homomorphisms from $G$ to $G'$. Let $H \subset G$ be the subset $\{x \in G: f(x)=g(x)\}$. Is $H$ a subgroup of $G$? Let $e$ and $e'$ ...
2
votes
1answer
33 views

Prove that if $f:G \longrightarrow G'$ is a group homomorphism, $f(x)^{-1} = f(x^{-1})$ for all $x \in G$

Can someone please verify this? Prove that if $f:G \longrightarrow G'$ is a group homomorphism, $f(x)^{-1} = f(x^{-1})$ for all $x \in G$ Let $e$ and $e'$ denote the identity elements of $G$ and ...
2
votes
0answers
45 views

Prove that the center of a group is normal.

Can someone please verify this? Prove that the center of a group is normal. let $G$ be a group, and let $\operatorname{Z}(G)$ denote its center. Let $g \in G$ and $z \in \operatorname{Z}(G)$. ...
0
votes
1answer
20 views

Prove that the composition of two group homomorphisms is a group homomorphism.

Prove that the composition of two group homomorphisms is a group homomorphism. Let $f:G \longrightarrow G'$ and $:G' \longrightarrow G''$ be two group homomorphisms. Let $x$ and $y$ be two ...
2
votes
1answer
32 views

Let $f:G \longrightarrow G'$ be a group homomorphism. Prove that $\operatorname{Im}(f)$ is a subgroup of $G'$.

Can someone please verify this? Let $f:G \longrightarrow G'$ be a group homomorphism. Prove that $\operatorname{Im}(f)$ is a subgroup of $G'$. Let $a,b \in \operatorname{Im}(f)$. Then, there ...
2
votes
1answer
56 views

Which of the following groups are subgroups?

I've written an answer for an exercise in Artin's algebra. Can someone please verify it? Which of the following groups are subgroups? (a) $GL_n(\mathbb{R}) \subset GL_n(\mathbb{C})$ (b) ...
4
votes
5answers
332 views

Prove any subgroup of a cyclic group is cyclic.

was just wondering if this is a valid proof for the aforementioned question? I am quite confident that it isn't, but not exactly sure why. Maybe I am missing the point of proofs by induction ...
2
votes
3answers
136 views

Group Homomorphism Questions (my attempts shown)

(a) Let $p$ be a prime. Determine the number of homomorphisms from $\Bbb Z_p \oplus \Bbb Z_p$ into $\Bbb Z_p$. Attempt: Suppose $\Psi:Z_p \oplus Z_p \rightarrow Z_p$ is an into homomorphism. ...
0
votes
0answers
55 views

Is my solution correct? Finite abelian groups are CLT groups.

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Use Cauchy's Theorem and induction to show that a finite abelian group has a subgroup of order $n$ for each ...
1
vote
1answer
36 views

$\hom_{\mathbb{Z}}(\mathbb{Z}_n, G) \cong G[n]$

I'm doing this exercise: G is an abelian group, prove that $$\hom_{\mathbb{Z}}(\mathbb{Z}_n, G) \cong G[n]= \lbrace g \in G | ng = 0 \rbrace$$ My attempt: Let's consider the exact sequence $$ 0 \to ...
0
votes
1answer
39 views

$MN/M \cap N \cong (MN/M) \times (MN/ N )$

I want to prove the following exercise from Dummit & Foote's Abstract Algebra: Let $M$ and $N$ be normal subgroups of $G$ such that $G=MN$. Prove that $G/M \cap N \cong (G/M) \times (G/N).$ ...
1
vote
0answers
34 views

Any non-trivial finitely-generated group admits maximal subgroups

I want to solve the following problem from Dummit & Foote's Abstract Algebra: This is exercise involving Zorn's Lemma (see Appendix I) to prove that every nontrivial finitely generated group ...
3
votes
2answers
79 views

False proof? If $f: G \rightarrow H$ homomorphism, and $H$ abelian, then $G$ abelian? [duplicate]

Let $f: G \rightarrow H$ be homomorphism, and $H$ is abelian. So $G \big/ \ker f \cong \operatorname{im}f$. Since $\operatorname{im}f$ is abelian, so is $G \big/ \ker f$. So for every $g_1,g_2 \in ...
3
votes
1answer
50 views

Cayley's Theorem - Questions on Proof Blueprint [Fraleigh p. 82 theorem 8.16]

Not a duplicate of this exquisite answer, the numbers in which I abide by here. Not querying the proof, hence please don't discourse on it. Proof blueprint: Steps 1-2 in words. Left multiplication ...
2
votes
0answers
41 views

Proof verification that group elements follow law of exponents

I have a proof for the following proposition: Suppose $G$ is a group, $g\in G$ and $m,n \in \mathbb{Z}$. Then $(g^m)^n=g^{mn}$ Proof If $m$ and $n$ are positive then it is clear that ...
4
votes
0answers
67 views

Proof “correctness” : Cycle structure of conjugate permutations

My Algebra lecturer is a very strict about proofs(w.r.t Completeness , correctness and format ) more so than I have encountered in the past or any of my lecturers of the courses I am take concurrent. ...
1
vote
0answers
38 views

proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian

If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian. Is that statement correct? Here's my attempt of proof: Let $a,b \in G$, then: ...
1
vote
1answer
36 views

Lang's proof of Cauchy's Theorem

In proving Cauchy's theorem in his 'Algebra', Lang first prove[s] by induction that if $G$ has exponent $n$ then the order of $G$ divides some power of $n$. Let $b \in G, b \ne 1$, and let $H$ be ...
1
vote
2answers
49 views

Proof of Conjugate Subgroup Isomorphism

Let $G$ be a group, and let $H$ be a subgroup of $G$. Prove that if $a$ is an element of $G$, then the subset $aHa^{-1} = \{g ∈ G | g = aha^-1 \text{ for some } h \in H\}$ is a subgroup of $G$ that is ...
1
vote
0answers
43 views

Verifying $G*H$ Has Trivial Center and Elements of Infinite Order

Hypothesis: Let $G \ne H$ denote two non-trivial groups. Goal: Show that $G * H$ has a trivial center (hence is non-abelian) and contains an element of infinite order. Is my attempted proof below ...
3
votes
0answers
34 views

Verifying $G*H \cong G' * H' \implies |G| = |G'|$ or $|G| = |H'|$ (All Groups Cyclic)

Hypothesis: Let $G$, $H$, $G'$, and $H'$ be cyclic groups of orders $m$, $n$, $m'$, and $n'$ respectively. Goal: Show that if $G * H$ is isomorphic to $G' * H'$ then $m = m'$ and $n=n'$ or else $m ...
0
votes
0answers
48 views

Describe the kernel and the fibers of $\phi$ geometrically (as subsets of the plane).

Define $\phi : \mathbb{C}^{\times} \mapsto \mathbb{R}^{\times}$ by $\phi(a+bi) = a^2 + b^2$. Prove that $\phi$ is a homomorphism and find the image of $\phi$. Describe the kernel and the fibers of ...
1
vote
1answer
41 views

Why is $H_1 \le G \land H_2 \le G$ necessary in $a(H_1 \cap H_2) = aH_1 \cap aH_2$?

The problem is as follows: $G$ is a group and $H_1$ and $H_2$ are its two subgroups (i.e., $H_1 \le G \land H_2 \le G$). To prove that $a(H_1 \cap H_2) = aH_1 \cap aH_2$. Here is my trial: On ...
0
votes
1answer
154 views

Proving that the unities of a ring form a group under multiplication

I am presented with the following task: Show that if $U$ is the collection of all units in a ring $\langle R, +, \cdot\rangle$ with unity, then $\langle U, \cdot\rangle$ is a group. I am still ...
3
votes
1answer
68 views

Describe the fibers of $\phi$ and that $\phi$ is a homomorphism.

Define $\phi: \mathbb{R}^x \mapsto \{\pm1\}$ by letting $\phi(x)$ be $x$ divided by the absolute value of $x$. Describe the fibers of $\phi$ and that $\phi$ is a homomorphism. Need help getting a ...
0
votes
0answers
36 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
58 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' ...
2
votes
2answers
97 views

Direct product of two normal subgroups

Let $A$ and $B$ be normal subgroups of a group $G$ such that $A \cap B = \langle e \rangle$ and $AB = G$. Prove that $A \times B \cong G$ Attempted proof: Define $f : A \times B \rightarrow G$ ...