Linked Questions

8 votes
1 answer
21k views

Union of subgroups is a subgroup if and only if one subgroup is a subset of the other [duplicate]

Let $H$ and $K$ denote two subgroups of a group $G$. Prove that the union $H \cup K$ is a subgroup of $G$ if and only if $H \subset K$ or $K \subset H$.
  • 91
3 votes
1 answer
5k views

Show that a group can not be expressed as union of two of its proper subgroups [duplicate]

Show that a group can not be expressed as union of two of its proper subgroups. I am not sure how to start.
  • 2,071
0 votes
1 answer
1k views

No group can be a union of two proper subgroups [duplicate]

I am trying to prove the following result. No group $G$ can be the union of two proper subgroups. The first point of confusion is that I have two different definitions of proper subgroup. My ...
  • 1,305
3 votes
1 answer
822 views

Group equals union of two subgroups [duplicate]

Suppose $G=H\cup K$, where $H$ and $K$ are subgroups. Show that either $H=G$ or $K=G$. What I did: For finite $G$, if $H\neq G$ and $K\neq G$, then $|H|,|K|\le |G|/2$. But they clearly share the ...
  • 3,225
1 vote
0 answers
47 views

Union subgroup implies that the group is one of the subgroups. [duplicate]

Suppose $G$ is a group and $H_1, H_2\leq G$ such that $H_1\cup H_2=G.$ How can I prove that either $G=H_1$ or $G=H_2$?
  • 11
4 votes
5 answers
1k views

If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$? [closed]

$P$ and $Q$ are subgroups of a group $G$. How can we prove that $P\cap Q$ is a subgroup of $G$? Is $P \cup Q$ a subgroup of $G$? Reference: Fraleigh p. 59 Question 5.54 in A First Course in Abstract ...
  • 451
3 votes
3 answers
438 views

If a union of two subgroups of $G$ equals $G$, must one of the subgroups equal $G$?

If the union of two subgroups of $G$ is the group $G$, does that mean one subgroup is $G$? In a problem, it was proven to be true on this site. But take real numbers under addition as a group with ...
  • 2,341
2 votes
1 answer
2k views

Tricks to Prove Union of Two Subgroups iff One is Contained in the Other - Fraleigh p. 54 - based on Exercise 5.4.5

Not a duplicate because I'm asking about tricks and the blueprint for the proof based on this. Let $H, K \le G$. Prove $H \cup K$ is a subgroup $\iff H \subseteq K$ or $K \subseteq H$. Backward ...
user avatar
4 votes
1 answer
8k views

Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces of $v$ is contained in the other. [duplicate]

Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces of $v$ is contained in the other. May someone please validate this proof: Let $V_1,V_2$ be two ...
5 votes
1 answer
863 views

A group with two non trivial subgroups is cyclic

Let $G$ be a group. Suppose that $G$ has at most two nontrivial subgroups. Show that $G$ is cyclic. Can anyone help me please to solve the problem?
  • 279
7 votes
1 answer
312 views

Group which can be written as a union of proper normal subgroups

Let $G$ be a group and $\{N_j\}_{j \in J} $ be a family of proper normal subgroups of $G$ such that $G=\cup_{j \in J} N_j$ and $N_i \cap N_j =\{e\}$ for every $i\ne j \in J$ . Then how to prove that ...
  • 1,423
3 votes
2 answers
181 views

A conditioned morphism in a group

Let $(G,+)$ be an abelian group of at least $3$ elements and $f:G \to G$ a homomorphism such that $f(x) \in \{0, x, -x\}$ for all $x \in G$. Show that $f \in \{0_G, 1_G, -1_G\}$. I tried proving that $...
0 votes
3 answers
454 views

Prove/disprove: $I \cup J$ is (always) an Ideal of $R$.

Let $I$ and $J$ be the ideals of $R$. Prove/disprove: $I \cup J$ is (always) an Ideal of $R$. Rough Sketch: Since, $I$ and $J$ are the ideals of $R$, we have $0_R \in I$ or $0_R \in J$. Hence, $0_R \...
  • 1,683
1 vote
3 answers
688 views

HK are subgroups of G

I'm proving that if $H$ is a subgroup of a group $G$ and $K$ is a normal subgroup of $G$, then $HK$ is a subgroup of $G$. I've tried pondering the fact that $HK$ is a subgroup of $G$ iff $HK = \...
  • 1,157
2 votes
2 answers
197 views

Characteristic of an integral domain: doubt in the proof

Suppose by absurd that the characteristic of an integral domain is an integer $n$ not prime, say $n=n_1 \cdot n_2$. Now we have $$na=(n_1 \cdot n_2)a=0 \implies (n_1 \cdot a) \cdot (n_2 \cdot a)=0 \...
  • 1,906

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