What’s on the Cyclic Groups poster
A definition and 7 facts worth remembering.
A cyclic group is generated by a single element: every member is a power of that generator.
- Written ⟨g⟩, where g is a generator
- Every cyclic group is abelian
- Infinite cyclic groups are isomorphic to ℤ
- Finite ones of order n are isomorphic to ℤₙ
- Every subgroup of a cyclic group is cyclic
- ℤₙ has exactly one subgroup per divisor of n
- ℤₙ has φ(n) generators, φ being Euler's totient
- Order of ⟨g⟩ — The order of g itself
- Generators of ℤₙ — φ(n) of them
- Subgroups of ℤₙ — One per divisor of n
- ℤ₆ has subgroups of order 1, 2, 3 and 6 — one for each divisor of 6.
A group of prime order is cyclic and has no proper nontrivial subgroup, by Lagrange. There is exactly one such group per prime.
Questions about the Cyclic Groups poster
- What’s on the Cyclic Groups poster?
- A definition and 7 facts worth remembering. A cyclic group is generated by a single element: every member is a power of that generator. Written ⟨g⟩, where g is a generator; Every cyclic group is abelian; Infinite cyclic groups are isomorphic to ℤ; Finite ones of order n are isomorphic to ℤₙ; Every subgroup of a cyclic group is cyclic; ℤₙ has exactly one subgroup per divisor of n; ℤₙ has φ(n) generators, φ being Euler's totient; Order of ⟨g⟩ — The order of g itself; Generators of ℤₙ — φ(n) of them; Subgroups of ℤₙ — One per divisor of n; ℤ₆ has subgroups of order 1, 2, 3 and 6 — one for each divisor of 6.. A group of prime order is cyclic and has no proper nontrivial subgroup, by Lagrange. There is exactly one such group per prime.
- Who is the Cyclic Groups poster for?
- Cyclic Groups belongs to the Mathematics section rather than to a school year, because math is not something one grade owns. Anyone learning abstract algebra can pin it up — a beginner, a student mid-course, or someone revising years later.
- When should you use the Cyclic Groups poster?
- Any group of prime order is cyclic, which settles a surprising number of problems. A wall chart earns its place by being glanceable from where the work is happening, so Cyclic Groups belongs on the wall where that math work actually happens, within glancing distance, rather than filed away.
- What other posters go with Cyclic Groups?
- Common Finite Groups, Cosets & Normal Subgroups and First Isomorphism Theorem sit alongside Cyclic Groups in the Mathematics section. Printed together they make a wall rather than a single sheet, which is how a reference set actually gets used.Common Finite GroupsCosets & Normal SubgroupsFirst Isomorphism Theorem
- Is the Cyclic Groups poster free to download and print?
- Yes. Cyclic Groups downloads as a free PDF with no account, no email and no watermark, like everything else in the Mathematics section. Print as many copies as you like for a home, a classroom, a library or a tutoring group; reselling the file is the only thing the licence rules out.Read the licence
- What size does the Cyclic Groups poster print at?
- Cyclic Groups is a vector PDF laid out for US Letter, and prints on A4 with Fit to page — the same file, no separate download. Because every mark on it is drawn rather than photographed, it stays sharp enlarged to A3, A2 or A1 at a copy shop. Colour carries emphasis only, so a greyscale print of Cyclic Groups loses nothing.Printing guide
Related posters
Charts that sit alongside Cyclic Groups on the same wall.
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