Lychnis alpina; the alpine-catchfly is common on dryer soils and gravel-rich grounds. It is a delicate pretty little plant which barely exceeds 15 cm but normally even shorter. Because it often grows on sparsely vegetated grounds it can still be quite conspicuous. Generally speaking this plantlet develops one or few flowerstalks on marginal grounds, but more on favourable soils. The flower color normally is pink tot purple but can be white too - but these variants have purple stems as well. I noticed that the white-flower colored variants are common only on peat and sandy soils of the east coast. I can be wrong here though. The purple variant is common throughout Iceland.
It is a member of the Pink family (Caryophyllaceae). The Icelandic name of this species is Ljósberi.