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Title: Subgroups
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Series: Algebra
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YouTube-Title: Algebra 10 | Subgroups
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Bright video: https://youtu.be/mqKxfhSsfZ4
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Dark video: https://youtu.be/PT05WluCFb4
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: alg10_sub_eng.srt missing
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $(G, \circ)$ be a group and $H \subseteq G$ be a subset. What is always correct?
A1: $H$ together with the binary operation $\circ$ forms a group.
A2: $H$ is non-empty.
A3: The neutral element of $(G, \circ)$ is an element of $H$.
A4: There is a group that contains $H$.
Q2: Let $(G, \circ)$ be a group and $H \subseteq G$ a subgroup, which means $(H, \circ)$ is also a group. What is always correct?
A1: $H$ has the same neutral element as $G$.
A2: $H$ has more than one element.
A3: $H$ has either one element or is $G$ itself.
A4: If $a \in G$, then $a^{-1} \in H$.
A5: The map $H \rightarrow G$ given by $x \mapsto x$ is not a group homomorphism.
Q3: Let $(G, \circ)$ be a group and $H \subseteq G$ be a non-empty subset. What is equivalent for $H$ being a subgroup?
A1: For all $a,b \in H$, we have $a \circ b \in H$ and $a^{-1} \in H$.
A2: For all $a,b \in H$, we have $a \circ b \in H$.
A3: For all $a,b \in H$, we have $a^{-1} \in H$.
A4: For all $a,b \in H$, we have $a \circ b \in H$ and $e_G \in H$.
A5: For all $a,b \in H$, we have $b \circ b \circ a \in H$.
Q4: Let $(G, \circ)$ be a group with more than two elements. What is always correct?
A1: $G$ has at least two subgroups.
A2: $G$ has at most two subgroups.
A3: $G$ has exactly two subgroups.
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Last update: 2024-11
