Linked Questions
20 questions linked to/from If a group is the union of two subgroups, is one subgroup the group itself?
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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 $K \subset H$.
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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.
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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 ...
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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 ...
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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 $G=H_2$?
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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 ...
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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 ...
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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 ...
user53259
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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 ...
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5
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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?
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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 ...
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3
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2
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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 $...
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3
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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 \...
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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 = \...
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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 \...
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