The Subgroup TestOne condition that replaces re-checking every group axiom.MathematicsA nonempty subset H of G is a subgroup exactlywhen ab⁻¹ lies in H for every a and b in H.H ≠ ∅ and a, b ∈ H ⇒ ab⁻¹ ∈ HEven integers inside ℤA subgroupClosed under a − b.Odd integers inside ℤNot a subgroupIt misses 0.Finite closed subsetA subgroupFinite gives inverses.THE COMMON MISTAKE✗ Checking closure under the operation only✓ Also checking inverses, or using ab⁻¹For an infinite subset closure alone is not enough: the positive integers areclosed under addition and contain no inverses at all.For a finite nonempty subset, closure under the operation is sufficienton its own. Infinite subsets need the inverse condition too.The Subgroup Testlearnposters.com
The Subgroup Test — printable math wall chart from LearnPosters. Free vector PDF, US Letter and A4.

What’s on the The Subgroup Test poster

The rule, 3 worked examples, and the mistake everyone makes.

For a finite nonempty subset, closure under the operation is sufficient on its own. Infinite subsets need the inverse condition too.

Questions about the The Subgroup Test poster

What’s on the The Subgroup Test poster?
The rule, 3 worked examples, and the mistake everyone makes. A nonempty subset H of G is a subgroup exactly when ab⁻¹ lies in H for every a and b…; H ≠ ∅ and a, b ∈ H ⇒ ab⁻¹ ∈ H; Even integers inside ℤ = A subgroup; Odd integers inside ℤ = Not a subgroup; Finite closed subset = A subgroup; Common mistake: Checking closure under the operation only → Also checking inverses, o…. For a finite nonempty subset, closure under the operation is sufficient on its own. Infinite subsets need the inverse condition too.
Who is the The Subgroup Test poster for?
The Subgroup Test 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 The Subgroup Test poster?
Whenever a subset of a known group is claimed to be a group in its own right. A wall chart earns its place by being glanceable from where the work is happening, so The Subgroup Test belongs on the wall where that math work actually happens, within glancing distance, rather than filed away.
What other posters go with The Subgroup Test?
Common Finite Groups, Cosets & Normal Subgroups and Cyclic Groups sit alongside The Subgroup Test 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 SubgroupsCyclic Groups
Is the The Subgroup Test poster free to download and print?
Yes. The Subgroup Test 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 The Subgroup Test poster print at?
The Subgroup Test 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 The Subgroup Test loses nothing.Printing guide

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