SmallGroup(p^4,3): Difference between revisions

Revision as of 16:02, 3 November 2009

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups

Definition

Let $p$ be a prime number. This group is defined by the followin presentation:

$G : = ⟨ a, b, c ∣ a^{p^{2}} = b^{p} = c^{p} = e, a b = b a, b c = c b, c a c^{- 1} = a b ⟩$

It is a group of order $p^{4}$ . Particular cases are $p = 2$ (we get SmallGroup(16,3) and $p = 3$ (we get SmallGroup(81,3)).

Revision as of 16:01, 3 November 2009 (view source) Vipul (talk \| contribs) (Created page with '{{prime-parametrized particular group}} ==Definition== Let <math>p</math> be a prime number. This group is defined by the followin presentation: <math>G := \langle a,b,c \…')		Revision as of 16:02, 3 November 2009 (view source) Vipul (talk \| contribs) (→‎Definition) Newer edit →
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	<math>G := \langle a,b,c \mid a^{p^2} = b^p = c^p = e, ab = ba, bc = cb, cac^{-1} = ab \rangle</math>		<math>G := \langle a,b,c \mid a^{p^2} = b^p = c^p = e, ab = ba, bc = cb, cac^{-1} = ab \rangle</math>

	It is a group of order <math>p^4</math>. Particular cases are <math>p = 2</math> (we get [[SmallGroup(16,3]]]) and <math>p = 3</math> (we get [[SmallGroup(81,3)]]).		It is a group of order <math>p^4</math>. Particular cases are <math>p = 2</math> (we get [[SmallGroup(16,3]]) and <math>p = 3</math> (we get [[SmallGroup(81,3)]]).